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For the purpose of classifying another algebraic structure which is parametrised by the choice of a group and of an automorphism I need the classification up to isomorphism of automorphism groups of p-groups of order $p^4$.

I duly searched the web for a while and all the group theory manuals I could lay my hands on but I didn't find anything, not a hint. Could someone provide a reference?

I already posted the same question on maths stack exchange and Professor Holt guessed that there is no publication on the subject. I repost here to verify his guess thoroughly.

A good reference for the classification of groups of order $p^4$ would be useful too, since I found the following thesis On p-groups of low power order but a different reference to help me compare the possible approaches to the problem would be of great help!

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    $\begingroup$ It's not a complete answer to your question, but if you want to get some intuition, GAP (and hence SAGE) has a library of all groups of order $p^4$ built in. See gap-system.org/Packages/sgl.html $\endgroup$
    – Russ Woodroofe
    Commented Feb 9, 2014 at 0:25
  • $\begingroup$ For $p=2$, si Hall-Senior, Atlas of groups of order $\le2^6$. $\endgroup$
    – yakov
    Commented Jun 28, 2016 at 22:01
  • $\begingroup$ Since it's not mentioned: for $p\ge 5$ there is (Malcev-Lazard) an equivalence of category between groups of order $p^4$ and Lie algebras of order $p^4$ over $\mathbf{Z}/p^4\mathbf{Z}$. So in a certain sense the classification is uniform for $p\ge 5$ (uniform doesn't mean the number of types depends on $p$ as it can involve parameters in $\mathbf{Z}/p\mathbf{Z}$, but the description is uniform). So the automorphism groups should be described in a uniform way too. $\endgroup$
    – YCor
    Commented Apr 25, 2020 at 8:23

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Here's a description of the groups of order $p^4$. It's kind of primitive and explicit, but it's not too long.

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    $\begingroup$ Unfortunately the link no longer works, and Marcel Wild's page at math.sun.ac.za/~mwild no longer exists. Can you please update this link that presumably went to your preprint "Groups of order $p^4$ made less difficult" with Garlow and Wheland? $\endgroup$
    – Noam D. Elkies
    Commented Jul 20, 2016 at 6:31
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    $\begingroup$ Sorry for being slow. I've been away from this forum, and only just saw your request. I posted the note on the arXiv, and fixed the link above to point there. $\endgroup$
    – Jeff Adler
    Commented Nov 3, 2016 at 1:32
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There is a description of the automorphism groups of groups of order $p^4$ in the Thesis by B. Girnat: "Klassifikation der Gruppen bis zur Ordnung $p^5$." Staatsexamensarbeit, TU Braunschweig, 2003, whose advisor was B. Eick. The description is given in terms of the action on a set of generators of the respective group. If I may add a personal note, I have been searching for this information for a long time, and I found it incredible that such a basic and old topic had no treatment whatsoever published in any form on the web. But I was assuming that anything interesting can be found in English (either written or translated into it). This assumption has been proven wrong, and I state this here as a warning for the newbies like me!

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Well, the simplest reference is groupprops wiki. This paper by Behravesh and Mousavi proves some results, but for the classification refers to the recent work by Burnside (1897), so you might want to look there.

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This has the content of a comment, but is somewhat longer and with more formatting than a comment allows, so I post it as an answer.

Anyway: I tried the GAP computation for 2^4, and got results. For 3^4, it already runs out of memory with the default settings. It seems likely that pumping up the amount of memory available (with the command line option -o) might allow the computation to finish for $p$=3 and perhaps 5, but it doesn't look like you'll be able to calculate everything for a large number of $p$'s.

The StructureDescription command I'm using below is also fairly inefficient, and there might be better ways to understand the automorphism groups.

For $p$=2, the command and output are:

List(AllSmallGroups(Size, 2^4), x->StructureDescription(AutomorphismGroup(x)));

[ "C4 x C2", "(C2 x C2 x A4) : C2", "(C2 x C2 x C2 x C2) : C2", "(C2 x C2 x C2 x C2) : C2", "C2 x D8", "C2 x D8", "(C2 x D8) : C2", "C2 x D8", "(C2 x D8) : C2", "(((C2 x D8) : C2) : C3) : C2", "(((C4 x C2) : C2) : C2) : C2", "(((C2 x C2 x C2 x C2) : C3) : C2) : C2", "C2 x S4", "A8" ]

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  • $\begingroup$ I'd add that thanks to prof. @russ-woodroofe reference to GAP I discovered that there is a specific package to compute automorphism groups of p-groups based on the p-group generation algorithm by Eamonn O'Brien. $\endgroup$
    – Giuliano Bianco
    Commented Feb 10, 2014 at 7:18
  • $\begingroup$ Magma is returning the orders of the automorphism groups for reasonable sized primes almost instantly. But I think it is using the Eick/O'Brien package, so perhaps you need to tell GAP to use it. Example: [#AutomorphismGroup(SmallGroup(13^4,i)): i in [1..NumberOfSmallGroups(13^4)] ]; [ 26364, 748526688, 53466192, 4455516, 4112784, 342732, 53466192, 4455516, 685464, 685464, 116770163328, 116770163328, 695060496, 57578976, 610296923230525440 ] $\endgroup$
    – Derek Holt
    Commented Feb 10, 2014 at 12:20
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    $\begingroup$ I tried this in GAP 4.7.2 (actually, in a more recent developer version, but it shouldn't matter). With the autpgrp loaded in GAP (which, by the way, is normally loaded by default when you have it installed), computations are basically instant for e.g. 13^4, too. $\endgroup$
    – Max Horn
    Commented Feb 10, 2014 at 14:27
  • $\begingroup$ Ah, however, computing StructureDescription on each of the automorphism groups won't be instant. That's because of how StructureDescription works. But StructureDescription is not a very useful tool to begin with. Read its documentation for a lot of warnings about its limitations; it certainly isn't the right tool for a "classification" of a set of groups. $\endgroup$
    – Max Horn
    Commented Feb 10, 2014 at 14:29
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    $\begingroup$ The reason for the inefficiency of StructureDescription is mainly that originally there was the promise that the result should only depend on the isomorphism type of the group, and not on the way it is represented. Now as that promise in the GAP manual has been abandoned, it would be possible to speed up the function quite a lot -- of course at the 'cost' that the descriptions of some groups change. (I happen to be the author of a large part of StructureDescription.) $\endgroup$
    – Stefan Kohl ♦
    Commented Feb 10, 2014 at 18:43

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