Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#13020 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1282 conjugacy classes of subgroups.
| minimal degree | μ(G) = 70 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 70 points:
(1,2);(4,6)(7,8,13,20,37,53,38,19,15,22,11,18,14,21,12,39)(9,23,27,46,41,44,26,16,29,45,25,55,28,52,40,47)(10,30,33,51,42,48,36,50,35,54,31,17,34,49,32,56)(24,58,61,57,65,70,60,66,63,69,59,67,62,43,64,68);(3,4,5,6)(7,9)(8,16)(10,24)(11,26)(12,27)(13,40)(14,41)(15,29)(17,43)(18,52)(19,23)(20,44)(21,55)(22,47)(25,38)(28,37)(30,66)(31,60)(32,61)(33,64)(34,65)(35,63)(36,59)(39,45)(42,62)(46,53)(48,57)(49,67)(50,58)(51,70)(54,68)(56,69);(7,10,15,35)(8,17,22,51)(9,24,29,63)(11,36,13,32)(12,33,38,31)(14,42,37,34)(16,43,47,70)(18,30,20,54)(19,48,39,49)(21,56,53,50)(23,57,45,67)(25,60,27,64)(26,59,40,61)(28,65,41,62)(44,68,52,66)(46,58,55,69)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401712);