Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#33458 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator | C8 | C24×C4 |
| Frattini, cosocle | C2×C8 | C25 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 70 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 70 points:
(1,2);(3,4,6,5)(7,8)(9,19)(10,15)(11,27)(12,32)(13,16)(14,21)(17,30)(18,29)(20,44)(22,51)(23,45)(24,39)(25,54)(26,57)(28,42)(31,43)(33,41)(34,47)(35,50)(36,59)(37,56)(38,53)(40,49)(46,66)(48,69)(52,65)(55,64)(58,62)(60,70)(61,67)(63,68);(7,9,13,23)(8,14,16,35)(10,20,24,46)(11,28,31,59)(12,33,29,22)(15,34,39,61)(17,41,40,51)(18,42,32,36)(19,43,45,27)(21,49,50,30)(25,55,52,70)(26,58,53,48)(37,62,63,69)(38,64,57,60)(44,65,66,54)(47,68,67,56);(3,5,6,4)(7,10,13,24)(8,15,16,39)(9,20,23,46)(11,25,31,52)(12,26,29,53)(14,34,35,61)(17,37,40,63)(18,38,32,57)(19,44,45,66)(21,47,50,67)(22,48,33,58)(27,54,43,65)(28,55,59,70)(30,56,49,68)(36,60,42,64)(41,62,51,69);(8,16)(10,24)(14,35)(17,40)(18,32)(19,45)(20,46)(25,52)(26,53)(27,43)(36,42)(41,51)(47,67)(48,58)(55,70)(56,68)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529804);