TITLE: Is the classification of finite p-groups a smooth problem?
QUESTION [7 upvotes]: Fix a prime $p$. Then the question is about the difficulty of classifying finite $p$-groups. Originally I was going to ask if this was a "wild" problem but thanks to Joel David Hamkins' answer to question
10481 I now know to ask if it is a smooth problem.
At one time there was a minor industry producing papers which showed that a particular list of invariants did not classify finite $p$-groups so my understanding is that it would be remarkable if this problem was smooth.
Also for each prime $p$ and each $n$ the number of groups of order $p^n$ (up to isomorphism) is finite. This gives a sequence of integers for each $p$. For $p=2,3,5,7$ these sequences appear in OEIS as sequences A000679, A090091, A090130, A090140.
A supplementary question is:
Does knowing if the classification is or is not smooth have any bearing on the complexity of these sequences?

REPLY [3 votes]: 1) The classification of p-groups (even p-groups of class 2) is indeed wild over $\mathbb{F}_p$ [V. Sergeichuk, The classification of metabelian p-groups (Russian), Matrix problems, Akad.
Nauk Ukrain. SSR Inst. Mat., Kiev, 1977, pp. 150-161.]
(It's not clear to me why Hamkins's answer on your other question led you away from asking about wildness...)
2) Rather than smoothness - since as pointed out by others, classification problems on finite spaces are trivially smooth - in addition to wildness, you might consider computational complexity as a measure of difficulty of a finite classification problem. In particular, the existence of a polynomial-time algorithm for a classification problem is some evidence that the classification is "easy." 
There is currently no known poly-time algorithm to test isomorphism of finite groups, given by their multiplication tables, and it is widely believed that p-groups (even those of class 2) are the hardest cases. (Beware of potential circularity in beliefs though: the wildness of classifying p-groups of class 2 is one of the pieces of evidence for the preceding belief, although there are others as well.)
Furthermore, despite all of the exciting recent progress on classifying groups by coclass, as far as I know this hasn't yet been used to make any progress towards a polynomial-time algorithm for testing isomorphism of p-groups. Furthermore, isomorphism of p-groups of class 2 are believed to be the hardest cases, and these are exactly the groups of maximal coclass, for which the recent classification results say the least.
(Polynomial-time as a version of "easiness" is a little subtle compared to mathematical experience: for example, the fact that finite simple groups are all generated by 2 elements (a consequence of CFSG) implies that there is a relatively simple polynomial-time algorithm to test isomorphism of finite simple groups (find a generating pair of elements in the one group, try all possible maps of that pair into the other group and check if each is an iso), but that algorithm relies on a massive, difficult theorem for its correctness. However, I still think the computational complexity of a classification problem is one useful gauge of its difficulty, particularly when it comes to finite problems.)