Finite Group DatabasebetaFinite Group Databasebeta| order | 20160 |
| isoclinism | isoclinic class of A8 |
| automorphism | Aut(G) = S8 (order 40320) · Inn(G) = G · Out(G) = C2 |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C2 |
| alternating | A8 |
| projective general linear | PGL(4,𝔽2) |
| projective special linear | PSL(4,𝔽2) |
| special linear | SL(4,𝔽2) |
| general linear | GL(4,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | G | C1 |
| LCS | G | |
| UCS | C1 | |
136 conjugacy classes of subgroups; only the 2 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 2 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:
(2,3,4,5,6,7,8);(1,2,3)14 conjugacy classes — view the full character table.
| maximal subgroup of | S8, C2×A8 |
| maximal normal subgroup of | S8, C2×A8 |
| maximal quotient of | C2×A8, #72227518 |
| automorphism group of | C24 |
| outer automorphism group of | C24 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3,4,5,6,7,8), (1,2,3));