Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C210 |
| direct product | C2×C8×#121 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C8 | C23 |
| commutator | C22 | C24×C8 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C23×C4 |
| prime radical | C22×#121 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4,5,6,7,8,9,10);(12,14)(16,18);(15,16,17,18);(11,12)(13,14)224 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#208650, C2×#210728, C2×C8×D42, … |
| maximal normal subgroup of | C2×#208650, C2×#210728, C2×C8×D42, … |
| maximal quotient of | C2×#210728, C2×#495798, #72285998, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279945);