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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.

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