Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11912 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#11914 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1530 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | #97 | C24 |
| prime radical | C24 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ #97 ▷ C2×C4 ▷ C2 ▷ C1 | |
697 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);(3,4,7,13,8,14,28,44,9,15,27,45,30,12,39,41)(5,16,18,47,19,10,31,33,20,34,52,42,29,35,25,58)(6,21,23,53,24,38,56,43,22,40,26,60,54,11,36,37)(17,32,49,62,50,61,59,65,48,63,51,64,57,46,55,66);(3,5,9,20)(4,10,15,35)(6,17,22,48)(7,25,27,31)(8,29,30,19)(11,32,38,63)(12,34,14,16)(13,42,45,47)(18,28,52,39)(21,46,40,61)(23,55,26,59)(24,57,54,50)(33,41,58,44)(36,49,56,51)(37,65,43,66)(53,62,60,64);(3,6,9,22)(4,11)(5,17,20,48)(7,26,27,23)(8,24,30,54)(10,32)(12,40)(13,43)(14,21)(15,38)(16,46)(18,51,52,49)(19,50,29,57)(25,59,31,55)(28,36,39,56)(33,64)(34,61)(35,63)(37,45)(41,53)(42,66)(44,60)(47,65)(58,62)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390209);