Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8674 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#8678 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #380 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,7)(6,8)(9,10)(11,12)(13,22)(14,27,29,34)(15,30)(16,20)(17,21)(18,24)(19,28,35,31)(23,32,39,42)(25,33)(26,37)(36,38,41,40);(4,7)(8,9)(11,13,15,24,17,26,16,25)(12,18,20,22,21,33,30,37)(14,23,28,38,29,39,31,40)(19,32,34,36,35,42,27,41);(4,8)(6,10)(7,9)(11,14)(12,19)(13,23)(15,28)(16,31)(17,29)(18,32)(20,34)(21,35)(22,36)(24,38)(25,40)(26,39)(27,30)(33,42)(37,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 96476);