Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11989 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#11990 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1551 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | C24 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
687 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);(3,4,7,13,8,12,26,38,9,15,27,41,29,14,42,40)(5,16,18,44,19,33,48,58,20,32,49,39,28,10,25,31)(6,21,23,50,24,11,34,35,22,37,51,63,52,36,60,62)(17,30,46,56,47,43,64,65,45,57,59,66,54,55,53,61);(3,5,9,20)(4,10,15,33)(6,17,22,45)(7,25,27,48)(8,28,29,19)(11,30,36,57)(12,32,14,16)(13,39,41,44)(18,26,49,42)(21,43,37,55)(23,53,51,64)(24,54,52,47)(31,38,58,40)(34,59,60,46)(35,61,62,65)(50,56,63,66);(3,6)(4,11,15,36)(5,17)(7,23)(8,24)(9,22)(10,30,33,57)(12,37,14,21)(13,35,41,62)(16,43,32,55)(18,46)(19,47)(20,45)(25,53)(26,34)(27,51)(28,54)(29,52)(31,56,58,66)(38,63,40,50)(39,61,44,65)(42,60)(48,64)(49,59)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390429);