gap-forum

Dear MORETO QUINTANA Alexander,

Before exploring large data base entries it pays to look at the
relevant information. In this case the information shows
(which I have marked with *) that only 6675 groups have p-class
5 or more. All the groups of order 512 with derived length 3 occur
among these (though not all of them have derived length 3).

Mike (Newman)

gap> 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 - 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.