Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#33588 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
2024 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);(4,6)(7,8,13,18)(9,14,20,35)(10,23,26,57)(11,27,31,56)(12,32,29,17)(15,38,40,24)(16,41,43,30)(19,45,46,68)(21,51,49,67)(22,54,50,37)(25,58,33,42)(28,44,59,39)(34,60,61,47)(36,64,63,53)(48,69,55,65)(52,66,70,62);(3,4,5,6)(7,9)(8,14)(10,19)(11,21)(12,22)(13,20)(15,34)(16,36)(17,37)(18,35)(23,45)(24,47)(25,48)(26,46)(27,51)(28,52)(29,50)(30,53)(31,49)(32,54)(33,55)(38,60)(39,62)(40,61)(41,64)(42,65)(43,63)(44,66)(56,67)(57,68)(58,69)(59,70);(7,10,13,26)(8,15,18,40)(9,19,20,46)(11,28,31,59)(12,33,29,25)(14,34,35,61)(16,42,43,58)(17,44,32,39)(21,52,49,70)(22,55,50,48)(23,56,57,27)(24,41,38,30)(36,65,63,69)(37,66,54,62)(45,67,68,51)(47,64,60,53);(9,20)(14,35)(19,46)(21,49)(22,50)(34,61)(36,63)(37,54)(45,68)(47,60)(48,55)(51,67)(52,70)(53,64)(62,66)(65,69)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530127);