Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1357 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
| direct product | C2×#7268 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #331 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C24 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2);(3,4,5,6)(8,12)(9,16)(10,18)(13,20)(14,15)(17,22);(7,8,9,13,10,12,16,19,11,14,17,20,18,15,22,21)92 conjugacy classes — view the full character table.
| maximal subgroup of | #72214381, #72215347, #72216704, … |
| maximal normal subgroup of | #72214381, #72215347, #72216704, … |
| maximal quotient of | C2×#93382, C4×#7268, D3×#7268 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34890);