Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#13213 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#493 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(4,6)(7,8,11,14,12,16,21,15)(9,17,13,20,19,22,18,10);(7,9,10,8,11,18,17,15,12,19,20,16,21,13,22,14);(3,4,5,6)104 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#496124, #72224820, #72224822, … |
| maximal normal subgroup of | C2×#496124, #72224820, #72224822, … |
| maximal quotient of | #72225285, #72225286, C2×#141754, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402333);