To what extent is there/can there be a description

that is uniform in p(for p sufficiently large) of the p-groups of order $p^n$, for each fixed n?

**Note 1:** I used the word "description" rather than "classification" because I understand that classifying p-groups is notoriously open and difficult. The word "description" is meant to be headed in the general direction of "classification," but somewhat more generous and general.

**Note 2:** I'm asking more about the *possibility* of such a uniform-in-$p$ description, and what it could/must/can't look like, rather than *just* asking about what kind of classification results are currently known or the difficulty of such a classification.

**Note 3 (added Feb 10, 2014 2:40pm EST)**: I changed it from "independent in $p$" to "*uniform* in $p$" because I think that better captures my intention. I don't mean completely independent of (large) p. As in example (2) and Derek Holt's answer, if there is some description of, say, groups of order $p^{12}$ that depends on, say, the residue class of $p$ mod 20 (a fixed number), and on the number of solutions mod $p$ of a fixed (set of) equation(s) over $\mathbb{Z}$, then I'd be happy with that. The hope is sort of that there should be a finite description, *uniform in $p$*; that one doesn't need a completely new description for infinitely many values of $p$.

**Motivating Examples**

1) One of the examples which motivated this question: there is a classification of p-groups of order $p^4$ (going back to Burnside, I believe) for p=2, p=3, and then - crucially for this question - *all $p \geq 5$*.

2) Another infinite family of partial examples (partial because it only concerns some and not all groups of order $p^n$ for some fixed $n$): isomorphism classes of p-groups of class 2, exponent p, with $G/[G,G]$ of rank $t$ and $[G,G]$ of rank $z$ are in bijective correspondence with orbits of $z$-dimensional spaces of antisymmetric $t \times t$ matrices under change of basis on $G/[G,G]$. I understand that any description of the orbits of the latter action may vary with p (see this related question). But I *think* I nonetheless want to count this description as "independent of p", since the change-of-basis group and the action can all be defined over $\mathbb{Z}$ and then just taken mod p to get the picture for a given p (if I understand correctly).

3) **Added Feb 10, 2:40pm EST**: Derek Holt pointed out examples of du Sautoy and Vaughan-Lee suggesting that a classification of groups of order $p^{10}$ should be incredibly difficult. Although I find the difficulty interesting, in this question I'm more interested in the logical/model-theoretic/algebro-geometric/what-have-you *possibility* of the *existence* of such a description. For the particular examples in Derek Holt's answer, they give a classification of some groups of order $p^{10}$ that depend only on the value of $p$ mod 12 and on the number of solutions mod $p$ of the pair of integer equations $x^4 + 6x^2 - 3 = y^2 - x^3 + x = 0$. Although this number can vary a lot with $p$, this is still a single, finite description that is uniform in $p$ (and a very interesting example!).

regular, and so they have type invariants that can help give a generic description. This is what leads Hall to say that "most" $p$-groups are regular (in the sense that for a fixed $n$, only those with $p<n$ may fail to be regular), and this is what underlies the fact that descriptions of groups order $p^n$ tend to break down into $p\geq n$ (the "small class" case) and then special analysis for the cases of $p<n$. – Arturo Magidin Feb 10 '14 at 17:45order, related to algorithms for group isomorphism. The hardest cases seem to be the p-groups groups of class 2, which are of maximal coclass. – Joshua Grochow Feb 10 '14 at 19:27