Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#13315 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1480 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);(3,4)(5,6)(7,8,14,20)(9,21,26,52)(10,27,31,49)(11,32,36,47)(12,19,13,18)(15,38,42,23)(16,37,45,30)(17,34,48,35)(22,53,55,70)(24,46,25,51)(28,50,57,68)(29,60,63,44)(33,40,56,41)(39,65,66,59)(43,61,67,54)(58,64,69,62);(7,9,14,26)(8,15,20,42)(10,28,31,57)(11,33,36,56)(12,25,13,24)(16,43,45,67)(17,46,48,51)(18,41,19,40)(21,47,52,32)(22,37,55,30)(23,34,38,35)(27,39,49,66)(29,61,63,54)(44,68,60,50)(53,69,70,58)(59,64,65,62);(4,6)(7,10)(8,16)(9,22)(11,29)(12,37)(13,30)(14,31)(15,39)(17,44)(18,49)(19,27)(20,45)(21,50)(23,54)(24,57)(25,28)(26,55)(32,58)(33,59)(34,64)(35,62)(36,63)(38,61)(40,67)(41,43)(42,66)(46,53)(47,69)(48,60)(51,70)(52,68)(56,65)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406001);