Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#13525 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #482 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | #2092 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
726 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);(3,4,7,13,32,48,8,14,10,16,34,50,9,15,33,49)(5,17,19,39,21,38,20,41,23,51,35,40,22,46,30,11)(6,24,26,12,36,43,31,52,29,47,61,45,28,59,27,44)(18,42,54,53,25,60,58,66,57,65,62,64,56,37,55,63);(3,5,10,23)(4,11,16,41)(6,25)(7,30,34,20)(8,35,33,19)(9,21,32,22)(12,42)(13,46,50,38)(14,51,49,17)(15,39,48,40)(18,36)(24,53)(26,54)(27,58)(28,57)(29,56)(31,55)(37,52)(43,63)(44,60)(45,65)(47,64)(59,66)(61,62);(3,6)(4,12)(5,18)(7,31)(8,26)(9,36)(10,29)(11,37)(13,47)(14,43)(15,52)(16,45)(17,53)(19,58)(20,54)(21,56)(22,25)(23,57)(24,50)(27,34)(28,32)(30,62)(33,61)(35,55)(38,63)(39,65)(40,42)(41,60)(44,48)(46,66)(49,59)(51,64)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406715);