Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32929 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
| direct product | C2×#33171 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4430 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);(4,6)(7,8)(9,10)(12,15)(13,16)(17,18);(11,12,16,18,14,17,13,15);(3,4,5,6);(8,10)152 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D8×#121, C2×#6578155, C2×#6630311, … |
| maximal normal subgroup of | C2×D8×#121, C2×#6578155, C2×#6630311, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6537208);