Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#7545 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | D32 |
| commutator | C32 | C22×C4 |
| Frattini, cosocle | C2×C32 | C23 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 70 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 70 points:
(1,2);(3,4,5,6)(8,14)(9,15)(10,20)(11,24)(12,27)(16,38)(17,42)(18,45)(19,33)(21,49)(22,52)(23,37)(25,54)(26,48)(28,53)(29,30)(31,39)(32,43)(34,35)(36,50)(40,67)(41,65)(44,68)(46,47)(51,70)(55,64)(56,63)(57,59)(58,69)(60,66)(61,62);(7,8,15,30,10,16,34,42,24,39,49,64,47,56,22,32,12,18,36,58,25,40,51,60,28,44,62,65,48,57,23,33,13,19,37,59,26,41,61,68,53,66,70,67,54,69,50,45,27,43,52,63,46,55,21,31,11,17,35,38,20,29,9,14)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60767);