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×#13016 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | #1879 |
| prime radical | C22×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1288 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);(3,4,9,15,10,16,32,48,11,17,8,33,31,47,7,14)(5,18,22,41,23,46,20,49,24,52,21,12,36,39,34,40)(6,25,27,50,28,61,26,13,29,43,30,44,37,42,35,45)(19,53,55,60,56,38,58,51,57,64,62,66,59,65,54,63);(3,5,10,36,11,24,31,23)(4,12,16,49,17,41,47,40)(6,19,37,56,29,57,28,59)(7,21,9,20,32,22,8,34)(13,38,50,64,45,65,44,53)(14,46,15,18,48,39,33,52)(25,60,42,51,43,66,61,63)(26,55,27,58,35,62,30,54);(3,6)(4,13)(5,19)(7,30)(8,35)(9,26)(10,37)(11,29)(12,38)(14,25)(15,42)(16,50)(17,45)(18,51)(20,55)(21,54)(22,58)(23,59)(24,57)(27,32)(28,31)(33,61)(34,62)(36,56)(39,66)(40,53)(41,65)(43,48)(44,47)(46,60)(49,64)(52,63)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401707);