Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33758 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33763 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12)(13,22)(14,19)(15,27)(16,31)(17,21)(18,25)(20,29)(23,38)(24,39)(26,37)(28,34)(30,41)(32,36)(33,40)(35,42);(3,5)(4,7)(6,10)(8,9)(13,23)(18,33)(22,38)(24,32)(25,40)(28,42)(34,35)(36,39);(5,10)(7,9)(11,13)(12,18)(14,24)(15,28)(16,32)(17,23)(19,34)(20,36)(21,33)(22,27)(25,29)(26,40)(30,42)(31,35)(37,39)(38,41);(4,8)(7,9)(11,14,17,16)(12,19,21,31)(13,24,23,32)(15,26,30,29)(18,34,33,35)(20,27,37,41)(22,39,38,36)(25,28,40,42)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530491);