Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C4×#2195 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#447 |
| prime radical | C2×#578 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2,3,4);(5,6,12,18)(7,19,23,47)(8,24,27,44)(9,28,32,42)(10,17,11,16)(13,35,38,21)(14,33,40,26)(15,30,43,31)(20,48,52,65)(22,53,34,41)(25,56,59,39)(29,45,60,37)(36,57,64,68)(46,51,67,50)(49,54,62,63)(55,61,66,58);(5,7)(6,13)(8,20)(9,29)(10,22)(11,34)(12,23)(14,36)(15,41)(16,37)(17,45)(18,38)(19,28)(21,31)(24,54)(25,57)(26,50)(27,52)(30,35)(32,60)(33,51)(39,65)(40,64)(42,47)(43,53)(44,63)(46,55)(48,56)(49,58)(59,68)(61,62)(66,67);(5,8,12,27)(6,14,18,40)(7,20,23,52)(9,25,32,59)(10,33,11,26)(13,36,38,64)(15,39,43,56)(16,44,17,24)(19,46,47,67)(21,49,35,62)(22,51,34,50)(28,55,42,66)(29,57,60,68)(30,61,31,58)(37,63,45,54)(41,65,53,48)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402910);