Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94192 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1391 |
| commutator | C22×C8 | C42 |
| Frattini, cosocle | #2960 | C22 |
| socle | C2 | #7248 |
| prime radical | C23 | #338 |
| LCS | G ▷ C22×C8 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C22×C8 ◁ #2960 ◁ G | |
| LECS | G ▷ #2960 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
135 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,25,13,27,45,7,14,28,41,8,31,32,42)(3,15,17,47,52,48,24,57,19,35,54,61,20,55,56,64)(4,10,21,38,6,29,30,59,22,12,43,40,23,44,26,58)(9,33,34,50,36,60,63,51,37,53,62,16,46,49,39,18);(1,3,6,18,25,52,22,50,7,19,23,51,8,20,4,16)(2,9,12,35,31,46,29,48,14,37,10,15,13,36,44,55)(5,24,26,49,27,54,21,33,28,56,30,60,32,17,43,53)(11,39,38,64,42,62,58,61,41,63,40,57,45,34,59,47)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fgi,d^2=fg,e^2=fgh,g^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fh,g^a=gh,h^a=hi,d^b=dg,d^c=dh,e^d=ei,f^d=fi,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1461);