Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11704 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#11752 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(4,8,9,10)(11,12,15,20,17,21,16,31)(13,22,25,40,23,39,32,36)(14,26,28,37,30,38,29,19)(18,27,34,41,33,42,35,24);(3,4)(5,8)(6,10)(7,9)(11,13,17,23)(12,18,21,33)(14,27,30,42)(15,25,16,32)(19,36,37,40)(20,34,31,35)(22,38,39,26)(24,28,41,29);(3,5,7,6)(4,8,9,10)(13,23)(18,33)(22,39)(24,41)(25,32)(27,42)(34,35)(36,40)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388779);