Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13033 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#13035 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1274 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);(3,4,8,15,33,36,32,13,10,17,31,47,9,16,7,14)(5,18,22,11,34,38,20,48,24,51,58,40,23,45,21,39)(6,25,27,44,35,46,26,43,29,41,30,12,28,42,60,49)(19,37,54,62,59,65,53,61,56,64,57,50,55,63,52,66);(3,5,9,34,10,24,33,23)(4,11,36,39,17,40,16,48)(6,19,35,55,29,56,28,59)(7,22,31,20,32,58,8,21)(12,37,43,65,44,64,49,63)(13,45,47,51,14,38,15,18)(25,50,42,66,41,62,46,61)(26,57,30,53,60,54,27,52);(3,6)(4,12)(5,19)(7,30)(8,26)(9,35)(10,29)(11,37)(13,42)(14,46)(15,25)(16,49)(17,44)(18,50)(20,54)(21,57)(22,53)(23,59)(24,56)(27,32)(28,33)(31,60)(34,55)(36,43)(38,61)(39,65)(40,64)(41,47)(45,66)(48,63)(51,62)(52,58)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401750);