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.



How to generate all finite groups of order n?
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>.


