Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#13241 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C4×D8 |
| prime radical | C22×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(4,6)(7,8,13,19,14,21,30,20)(9,16,24,36,25,33,38,37)(10,26,22,15,28,34,23,35)(11,29,31,17,12,32,27,18);(3,4,5,6)(7,9,12,23,30,38,31,28,14,25,11,22,13,24,27,10)(8,15,17,16,19,34,32,36,21,35,18,33,20,26,29,37);(7,10,27,24,13,22,11,25,14,28,31,38,30,23,12,9)(8,16,32,35,20,37,17,34,21,33,29,15,19,36,18,26)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402418);