The number of groups of order 32 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.
 A: 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.
A: 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:

*

*Assume (by induction) groups of smaller order have been classified

*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.

*Compute the 2-cohomology group $H^2(G,M)$ and the corresponding extensions

*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.)

*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 $T\le G\le Aut(T)$.

A: 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!
