Is there anything significant about GAP's SmallGroup(512,2045)?

Question

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?

Answer 1

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 Sergey Malyushitsky (On Sylow $2$-subgroups of Finite Simple Groups of Order up to $2^{10}$, 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 $2$.

Answer 2

Besides the already accepted answer, another way of looking at this family of groups is as follows:

Let $V_d = (\mathbb Z/2)^d$. General theory says that there is a universal central extension $$ H_2(V_d;\mathbb Z/2) \rightarrow P_d \rightarrow V_d,$$ and the group in the question is $P_3$.

A lot is known about the cohomology rings of these groups (and their analogues at odd primes) by the work of Adem, Karagueuzian, and Minác in a 1999 paper On the cohomology of Galois groups determined by Witt rings in Advances, and I had fun with these also in a 2007 Advances paper Primitives and central detection numbers in group cohomology.