Number of non-Abelian groups of order $2^n$ Related to A000679 (Number of groups of order $2^n$), how many non-Abelian groups of order $2^n$ are there?
 A: Marshall Hall and James Senior published a book The Groups of order 2n (n <= 6)  in the 1960's.  It's in the common room in the math department here at UCSB.  Using rather obscure notation it arranges the conjugacy classes into lattices.
Online you can find a list of all groups of order 64.
A: It is as Qiaochu Yuan says, and worse: the number of abelian groups of order 2^n are not complicated. They are direct products of cyclic groups of order 2^m. The number of non-abelian groups is unknown. We don't actually have a formula. We just have exhaustive analyses for a few small values of n. 
A: It is true that there is no known formula for the number of isomorphism classes of groups of order $n$, but there is a very nice asymptotic formula for $p$-groups.
In particular, the number of isomorphism classes of groups of order $p^n$ grows as $p^{\frac{2}{27}n^3 + O(n^{8/3})}$.  This function grows very rapidly, and there is a folklore conjecture that 
"almost all groups are $2$-groups."
http://en.wikipedia.org/wiki/P-group
A: I don't think there's a general method of approaching this.There are several methods of obtaining all the Abelian groups of order n for specific n-such as the finite groups of prime order are all cyclic and therefore Abelian,etc. Unless n is very large,proceeding in this manner will usually get the number of groups down to a managable size and what remains should be non-Abelian.
I think that's the best you can do unless you want to add the condition that the remaining groups are simple-in which case,a Sylow analysis would be appropriate and would considerably simplify things.
There are a few general results on non-Abelian groups-like a finite group G is nonAbelian if there are 2 elements in G whose commutator is nontrivial i.e. not the identity. But I don't know if you can use these kinds of results to get the kind of general formula you want-I don't think you can,although I could be wrong on this.
The best discussions I know of such matters can be found in Herstien's  Topics in Algebra,2nd edition and I.Martin Issacs' Finite Group Theory. 
