The number of groups of order 32

Question

There are 51 groups of order ($32=2^5$). My question is how this number was computed.

Graham Higman and Charles Sims gives an estimate for the number of $p$-goups (i.e. groups of order $p^n$ where $p$-is a prime number).

I will be very grateful for any suggestion.

Answer 1

The very first entry in OEIS is the number of different groups of order $n$. Let us denote by $f(n)$ this number. There are estimates in general, as you said, but explicit numbers are impossible in general. H. U. Besche, B. Eick and E. A. O'Brien wrote a paper The groups of order at most $2000$, where $f(n)$ is determined for $1\le n\le 2000$. The powers of $2$ have the biggest numbers, e.g., $$ f(2^5)=51,\ldots ,f(2^{10})=49 487 365 422. $$ The paper answers how these numbers were computed.

Answer 2

The number of groups is computed by explicitly constructing all groups. This is done (originally by hand, nowadays on the computer -- see the paper by Besche/Eick/Obrien mentioned in Dietrich Burde's reply) by considering the ways groups of the given order can be formed as extensions. BAsically the steps are:

1. Assume (by induction) groups of smaller order have been classified
2. For each group $G$, classify the irreducible modules. For each module $M$ such that $|M|\cdot|G|$ has the right order. For prime powers only the trivial module needs to be considered.
3. Compute the 2-cohomology group $H^2(G,M)$ and the corresponding extensions
4. Eliminate further isomorphisms (i.e. isomorphisms that do not preserve the extension structure. (This is basically the costly part and the achilles heel of any such classification.)
5. Groups that have no solvable normal subgroup need to be handled separately (with simple groups as a seed). Essentially one needs to classify subgroups of wreath products $Aut(T)\wr S_m$ for $T$ simple and small $m$ (and subdirect products of these groups). For any plausible order bound this mosly degenerates to groups $T\le G\le Aut(T)$.

Answer 3

Just to put this discussion in context, the groups of order $32$ were first enumerated by G.A. Miller in the paper

Miller, G. A., The regular substitution groups whose orders is less than 48, , 28, (1896), 232–284,

which solves the problem for all order up to $48$. This was of course done by hand. Although plenty of mistakes were made in subsequent calculations of this type, as far as I know Miller's work is accurate. But unfortunately I don't have any easy access to the paper (the University of Warwick library does not go back that far).

The lists of groups of order $2^n$ for $n \le 6$ were published in a printed book in 1964:

Hall, Marshall, Jr.; Senior, James K. The groups of order $2^n\,(n\leq 6)$. The Macmillan Co., New York; Collier-Macmillan, Ltd., London 1964 225 pp.

I remember looking through that many years ago, but I expect the printed version is regarded as redundant now!