Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#26848 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2070 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)(5,10)(6,8)(7,9)(11,12)(13,24)(14,29)(15,33)(16,35)(17,22)(18,23)(19,27)(20,31)(21,34)(25,41)(26,40)(28,37)(30,42)(32,39)(36,38);(3,5,7,6)(4,8,9,10)(11,13,16,27,18,28,17,36)(12,19,22,24,23,38,35,37)(14,30,31,25,32,15,26,34)(20,33,29,41,40,42,39,21);(3,6,7,5)(4,8,9,10)(11,14,16,31,18,32,17,26)(12,20,22,29,23,40,35,39)(13,25,27,15,28,34,36,30)(19,21,24,33,38,41,37,42);(4,9)(8,10)(11,15,16,34,18,30,17,25)(12,21,22,33,23,41,35,42)(13,26,27,14,28,31,36,32)(19,20,24,29,38,40,37,39)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351149);