Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13490 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#13491 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2073 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
668 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);(3,4,10,31,7,13,9,15)(5,16,29,35,18,11,20,44)(6,12,23,39,24,37,30,45)(8,14,32,60,25,40,28,43)(17,46,49,64,50,33,53,62)(19,47,57,41,51,34,26,59)(21,54,52,65,55,36,56,63)(22,38,27,42,48,61,58,66);(3,5,7,18)(4,11,13,16)(6,21,24,55)(8,26,25,57)(9,29,10,20)(12,36,37,54)(14,41,40,59)(15,44,31,35)(17,48,50,22)(19,32,51,28)(23,56,30,52)(27,53,58,49)(33,61,46,38)(34,60,47,43)(39,65,45,63)(42,64,66,62);(3,6)(4,12,13,37)(5,17)(7,24)(8,27)(9,30)(10,23)(11,33,16,46)(14,42,40,66)(15,45,31,39)(18,50)(19,52)(20,53)(21,26)(22,28)(25,58)(29,49)(32,48)(34,63,47,65)(35,64,44,62)(36,59,54,41)(38,60,61,43)(51,56)(55,57)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406564);