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I know how to generate all Abelian groups of order n, but how would I generate the others? I can't seem to find anything about this.

By "generate", I mean produce the Cayley tables for all groups of order n.

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    $\begingroup$ There are electronic tables of such things. In general it's quite a hard problem. The last time I looked it wasn't even known how many groups of order $n$ there were for several $n$ less than 10000 (I might be out of date here, but I would feel pretty confident saying that the list of groups of order at most a million is almost certainly still not known). $\endgroup$
    – Kevin Buzzard
    Nov 21, 2010 at 20:20
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    $\begingroup$ What do you mean by "generate"? Other than "it's built out of simple groups of appropriate orders, but the extensions are a mess" or "write down every multiplication table and check that it gives a group", I don't think that there is going to be an easy answer. $\endgroup$
    – Andy Putman
    Nov 21, 2010 at 20:21
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    $\begingroup$ Here is an algorithm: Take all $n$-tuples of distinct elements in the symmetric group on $n$ letters (with one of them being identity), and check to see if they are closed under multiplication. For those that are, check all set-theoretic bijections between them to see if they are homomorphisms. This will give you a list of isomorphism class representatives, but it might take a while. $\endgroup$
    – S. Carnahan ♦
    Nov 22, 2010 at 2:46
  • $\begingroup$ Given the way in which $p$-groups proliferate as the power of $p$ goes up, there will always be practical constraints on what can be effectively computed for a given $n$. It may be more interesting to ask what can be learned by listing all groups of a given order. For example, early projects focused instead on finding all simple groups of order less than a million; this had a clear theoretical motivation. $\endgroup$
    – Jim Humphreys
    Nov 22, 2010 at 14:25
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    $\begingroup$ Unless the OP explains what s/he means by 'generate', this is not a real question. Voting to close. $\endgroup$
    – HJRW
    Nov 22, 2010 at 18:39

1 Answer 1

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See

Hans~Ulrich Besche, Bettina Eick, and E.A. O'Brien. A millennium project: constructing small groups. Internat. J. Algebra Comput., 12:623-644, 2002.

for a description of the construction of groups of order up to 2000. (I believe they narrowly failed to achieve this before the end of the year 2000.) In fact they did not construct the groups of order 1024 individually, but it is known that there are $49\,487\,365\,422$ groups of that order. The remaining $423\,164\,062$ groups of order up to 2000 (of which $408\,641\,062$ have order 1536) are available as libraries in GAP and Magma.

I would guess that 2048 is the smallest number such that the exact number of groups of that order is unknown. It is known that, for $p$ prime, the number of groups of order $p^n$ grows as $p^{\frac{2}{27}n^3+O(n^{8/3})}$: see http://en.wikipedia.org/wiki/P-group.

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    $\begingroup$ 1536? Gerhard "Ask Me About 512 Multiples" Paseman, 2010.11.21 $\endgroup$
    – Gerhard Paseman
    Nov 21, 2010 at 21:49
  • $\begingroup$ Sorry, yes 1512 should be 1536. $\endgroup$
    – Derek Holt
    Nov 22, 2010 at 9:46

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