Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#33525 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C22×#578 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
3822 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)(5,6)(7,8,13,18)(9,19,23,47)(10,15,24,38)(11,27,31,45)(12,32,29,17)(14,34,37,21)(16,41,43,30)(20,46,48,68)(22,51,33,42)(25,54,52,67)(26,57,53,40)(28,44,59,36)(35,60,61,49)(39,64,63,56)(50,69,58,65)(55,66,70,62);(7,9,13,23)(8,14,18,37)(10,20,24,48)(11,28,31,59)(12,33,29,22)(15,35,38,61)(16,42,43,51)(17,44,32,36)(19,45,47,27)(21,41,34,30)(25,55,52,70)(26,58,53,50)(39,65,63,69)(40,66,57,62)(46,67,68,54)(49,64,60,56);(4,6)(7,10)(8,15)(9,20)(11,25)(12,26)(13,24)(14,35)(16,39)(17,40)(18,38)(19,46)(21,49)(22,50)(23,48)(27,54)(28,55)(29,53)(30,56)(31,52)(32,57)(33,58)(34,60)(36,62)(37,61)(41,64)(42,65)(43,63)(44,66)(45,67)(47,68)(51,69)(59,70);(10,24)(15,38)(20,48)(25,52)(26,53)(35,61)(39,63)(40,57)(46,68)(49,60)(50,58)(54,67)(55,70)(56,64)(62,66)(65,69)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530012);