Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#2196 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C24 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
978 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 978 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 10 points:
(1,2);(3,4)(5,7)(6,8)(9,10);(4,7)(8,10);(5,9)40 conjugacy classes — view the full character table.
| maximal subgroup of | #152554, #153117, #153135, … |
| maximal normal subgroup of | #152554, #153117, #153135, … |
| maximal quotient of | C2×#12090, C2×#12096, C2×#12202, … |
| automorphism group of | #453, #454, #455, … |
| outer automorphism group of | C2×#329, #1525, #2979, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26531);