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 | C2×#12947 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | #1862 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1026 conjugacy classes of subgroups.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 66 points:
(1,2);(3,4,9,16,11,17,10,32)(5,18,22,48,23,49,33,36)(6,24,27,45,28,60,34,40)(7,15,29,44,30,46,8,14)(12,21,37,56,39,58,38,20)(13,25,41,62,43,31,42,26)(19,50,53,64,55,65,54,35)(47,57,63,61,66,51,59,52);(3,5,9,22,11,23,10,33)(4,12,16,37,17,39,32,38)(6,19,27,53,28,55,34,54)(7,21,29,56,30,58,8,20)(13,35,41,50,43,64,42,65)(14,36,15,18,44,48,46,49)(24,59,45,47,60,63,40,66)(25,61,62,51,31,52,26,57);(3,6)(4,13)(5,19)(7,26)(8,31)(9,27)(10,34)(11,28)(12,35)(14,24)(15,45)(16,41)(17,43)(18,47)(20,52)(21,57)(22,53)(23,55)(25,29)(30,62)(32,42)(33,54)(36,59)(37,50)(38,65)(39,64)(40,46)(44,60)(48,63)(49,66)(51,58)(56,61)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401512);