Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
| direct product | C2×#11618 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C22⋊C4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C2×#121 | C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
830 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);(3,4,7,10,8,5,6,9)(11,12,15,20,16,21,30,38,17,22,31,39,14,19,28,36)(13,23,26,40,25,33,41,37,24,35,29,42,27,18,32,34);(3,5)(4,8)(6,10)(7,9)(11,13)(12,18)(14,27)(15,29)(16,25)(17,24)(19,35)(20,37)(21,23)(22,33)(26,31)(28,41)(30,32)(34,39)(36,40)(38,42);(3,6,8,7)(4,9,5,10)(13,24)(18,33)(23,35)(25,27)(26,29)(32,41)(34,37)(40,42)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388297);