Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11669 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#11669 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #418 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | #1490 |
| prime radical | C2×#575 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
969 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4,6,13,9,14,18,12,10,15,7,17,8,5,16,11);(3,5)(4,9)(6,11)(7,12)(8,15)(10,14)(13,18)(16,17);(3,6,8,16,10,7,9,18)(4,11,14,13,15,12,5,17)92 conjugacy classes — view the full character table.
| maximal subgroup of | #72214259, #72214744, #72214826, … |
| maximal normal subgroup of | #72214259, #72214744, #72214826, … |
| maximal quotient of | #72214871, D3×#11669, C2×#128570, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388483);