gr.group theory - Counting the Groups of Order n Weighted by 1/|Aut(G)|

Computational evidence suggests $g(n)$ varies wildly with $n$. When $n$ is a power of a prime, or has lots of small factors, $g(n)$ can be very large (I would guess $g(2^k)$ is superexponential in $k$), and $a(n)$ contributes negligibly. In particular, for the prime power case, three-step nilpotent groups seem to dominate, and a good theoretical reason for this is that they asymptotically dominate in number of isomorphism types by a huge factor.

If $n$ has only a few factors, then $g(n)$ is a little bigger than $1/n$. In this case, $a(n)$ contributes nontrivially.

For those of you who are interested in groups of order $2^n$ (and who isn't?), I've computed a decomposition of $g(2^k), k leq 7$ by nilpotence class, so the class one columns on the left indicate $a(2^k)$, the abelian contribution.

g(2) = a(2) = 1.

nilpotence class |   1  
isom. types      |   1
weighted count   |   1

$g(4) = a(4) = 2/3 sim 0.67$.

nilpotence class |   1  
isom. types      |   2
weighted count   |  2/3

$g(8) = 23/42 sim 0.55$

nilpotence class |     1      |     2    
isom. types      |  3 (60%)   |  2 (40%) 
weighted count   | 8/21 (70%) |  1/6 (30%) 

$g(16) = 1247/2520 sim 0.49$

nilpotence class |       1      |     2     |   3
isom. types      |    5 (36%)   |  6 (43%)  |   3 (21%)
weighted count   | 64/315 (41%) | 1/6 (34%) | 1/8 (25%) 

$g(32) = 149297/312480 sim 0.48$

nilpotence class |        1        |       2      |      3     |   4
isom. types      |     7 (14%)     |   26 (51%)   |  15 (29%)  |  3 (6%)
weighted count   | 1024/9765 (22%) | 37/240 (32%) | 3/16 (39%) | 1/32 (7%) 

$g(64) = 48611383/78744960 sim 0.62$

$begin{array}{rccccc} text{nilpotence class} & 1 & 2 & 3 & 4 & 5 \ text{isomorphism types} & 11, (4%) & 117, (44%) & 114 , (43%) & 22, (8%) & 3, (1%) \ text{weighted count} & frac{32768}{615195} , (9%) & frac{3}{20} , (24%) & frac{5}{16} , (51%) & frac{3}{32} , (15%)& frac{1}{128} , (1%) end{array}$

$g(128) = 8999449693/8000487936 sim 1.12$

$begin{array}{rccccc} text{nilpotence class} & 1 & 2 & 3 & 4 & 5 & 6 \ text{isomorphism types} & 15 , (1%) & 947 , (41%) & 1137 , (49%) & 197 , (8%) & 29 , (1%) & 3 , (0%) \ text{weighted count} & frac{2097152}{78129765} , (2%) & frac{275929}{1451520} , (17%) & frac{2295}{3584} , (57%) & frac{7}{32} , (19%) & frac{3}{64} , (4%) & frac{1}{512}, (0%) end{array}$

The groups of order 64 took about 15 minutes on my ancient computer running GAP, and the groups of order 128 took about 40 hours. I'll leave the question of groups of order 256 to someone else, since a modern desktop computer should be able to work it out in a little under a year.

You might be curious about the three isomorphism types of nilpotence class $k-1$ for $k geq 4$. They are the dihedral, quasidihedral, and quaternion groups.

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