Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214×C4 |
| direct product | C23×#493 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
18754 conjugacy classes of subgroups; only the 4770 normal subgroups are shown.
This group has 18754 conjugacy classes of subgroups — too many to draw. Only its 4770 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 18 points:
(1,2);(3,4);(5,6);(8,10)(12,15)(13,16)(17,18);(11,12,16,18,14,17,13,15);(7,8,9,10)176 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#10304, C22×#21690, C2×#111526, … |
| maximal normal subgroup of | C22×#10304, C22×#21690, C2×#111526, … |
| maximal quotient of | C22×#23798, C2×#6357200, C22×#25779, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482456);