Finite Group DatabasebetaFinite Group Databasebeta| order | 40320 |
| automorphism | Aut(G) = S8 (order 40320) · Inn(G) = G · Out(G) = C1 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
| symmetric | S8 |
| socle, derived | A8 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, cosocle, hypocenter | A8 | C2 |
| LCS | G ▷ A8 | |
| UCS | C1 | |
296 conjugacy classes of subgroups; only the 3 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 3 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 8 |
| transitive degree | μt(G) = 8 |
A minimal faithful permutation representation — 2 generators on 8 points:
(1,2,3,4,5,6,7,8);(1,2)22 conjugacy classes — view the full character table.
| automorphism group of | A8, S8 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3,4,5,6,7,8), (1,2));