14
$\begingroup$

Here's the output of the GAP command "SmallGroupsInformation(512)"

There are 10494213 groups of order 512.

1 is cyclic. 
2 - 10 have rank 2 and p-class 3.
11 - 386 have rank 2 and p-class 4.
387 - 1698 have rank 2 and p-class 5.
1699 - 2008 have rank 2 and p-class 6.
2009 - 2039 have rank 2 and p-class 7.
2040 - 2044 have rank 2 and p-class 8.
2045 has rank 3 and p-class 2.
2046 - 29398 have rank 3 and p-class 3.
29399 - 30617 have rank 3 and p-class 4.
30618 - 31239 have rank 3 and p-class 3.
31240 - 56685 have rank 3 and p-class 4.
56686 - 60615 have rank 3 and p-class 5.
60616 - 60894 have rank 3 and p-class 6.
60895 - 60903 have rank 3 and p-class 7.
60904 - 67612 have rank 4 and p-class 2.
67613 - 387088 have rank 4 and p-class 3.
387089 - 419734 have rank 4 and p-class 4.
419735 - 420500 have rank 4 and p-class 5.
420501 - 420514 have rank 4 and p-class 6.
420515 - 6249623 have rank 5 and p-class 2.
6249624 - 7529606 have rank 5 and p-class 3.
7529607 - 7532374 have rank 5 and p-class 4.
7532375 - 7532392 have rank 5 and p-class 5.
7532393 - 10481221 have rank 6 and p-class 2.
10481222 - 10493038 have rank 6 and p-class 3.
10493039 - 10493061 have rank 6 and p-class 4.
10493062 - 10494173 have rank 7 and p-class 2.
10494174 - 10494200 have rank 7 and p-class 3.
10494201 - 10494212 have rank 8 and p-class 2.
10494213 is elementary abelian.

This size belongs to layer 7 of the SmallGroups library.

IdSmallGroup is not available for this size.

Even if I only barely know what "rank" and "p-class" are, 2045 on that list stands out. Is there any other way to describe it? For example, is it the Sylow subgroup of any simple group?

$\endgroup$
19
$\begingroup$

It is the "free" group of nilpotency class at most $2$, exponent dividing $4$, on three generators. (In other words, every group in that class is a quotient of it.)

According to

https://etd.ohiolink.edu/rws_etd/document/get/osu1086112148/inline

(ON SYLOW $2$-SUBGROUPS OF FINITE SIMPLE GROUPS OF ORDER UP TO $2^{10}$, Sergey Malyushitsky, M.S. Thesis, The Ohio State University)

it is not the Sylow $2$-subgroup of a finite simple group.

In general, the "free" group of nilpotency class at most $2$ and exponent at most $4$ on $d$ generators has order $4^d\cdot 2^{\binom{d}{2}}$. So, for example, you should see a similar phenomena at order 32, where there is a unique group of rank $2$ and $p$-class $2$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.