Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33758 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#33785 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1702 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12)(13,23)(14,22)(15,18)(16,29)(17,20)(19,26)(21,31)(24,40)(25,39)(27,34)(28,36)(30,38)(32,37)(33,41)(35,42);(3,5,6,10)(4,7,8,9)(12,18)(19,33)(20,22)(21,34)(23,39)(29,36)(35,38)(37,40);(5,10)(7,9)(11,13)(12,19)(14,24)(15,25)(16,30)(17,32)(18,33)(20,35)(21,37)(22,38)(23,29)(26,31)(27,41)(28,42)(34,40)(36,39);(4,8)(7,9)(11,14,15,17)(12,20,18,22)(13,24,25,32)(16,27,28,31)(19,35,33,38)(21,36,34,29)(23,37,39,40)(26,30,41,42)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530540);