Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#12942 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | #1862 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1026 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);(3,4,9,16,11,17,10,32)(5,18,22,36,23,49,33,45)(6,24,27,41,28,61,34,60)(7,15,30,46,31,47,8,14)(12,29,37,57,39,21,38,20)(13,40,42,62,44,25,43,26)(19,50,53,35,55,64,54,65)(48,58,66,56,63,51,59,52);(3,5,9,22,11,23,10,33)(4,12,16,37,17,39,32,38)(6,19,27,53,28,55,34,54)(7,29,30,57,31,21,8,20)(13,35,42,64,44,65,43,50)(14,45,15,18,46,36,47,49)(24,59,41,48,61,66,60,63)(25,56,26,51,40,52,62,58);(3,6)(4,13)(5,19)(7,26)(8,25)(9,27)(10,34)(11,28)(12,35)(14,24)(15,41)(16,42)(17,44)(18,48)(20,56)(21,58)(22,53)(23,55)(29,51)(30,40)(31,62)(32,43)(33,54)(36,66)(37,64)(38,50)(39,65)(45,59)(46,61)(47,60)(49,63)(52,57)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401499);