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×#13021 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | #1879 |
| prime radical | C2×#3414 | 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);(3,4,9,16,10,17,33,48,11,18,8,15,32,47,7,14)(5,19,22,41,23,46,21,51,24,49,31,12,36,39,34,40)(6,25,28,52,29,50,26,13,30,43,27,44,37,42,35,45)(20,54,56,61,57,38,62,63,58,64,59,66,60,65,55,53);(3,5,10,36,11,24,32,23)(4,12,17,51,18,41,47,40)(6,20,37,57,30,58,29,60)(7,31,9,21,33,22,8,34)(13,38,52,64,45,65,44,54)(14,46,16,19,48,39,15,49)(25,61,42,63,43,66,50,53)(26,56,28,62,35,59,27,55);(3,6)(4,13)(5,20)(7,28)(8,26)(9,35)(10,37)(11,30)(12,38)(14,43)(15,42)(16,50)(17,52)(18,45)(19,53)(21,59)(22,55)(23,60)(24,58)(25,48)(27,33)(29,32)(31,62)(34,56)(36,57)(39,61)(40,54)(41,65)(44,47)(46,66)(49,63)(51,64)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401718);