Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C82 · Schur multiplier (H2) = C23×C8 |
| direct product | C8×C22⋊C8 |
| unambiguous semidirect product | C22⋊C82 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C8 | C22 |
| commutator | C2 | C2×C82 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C23 | C2×C42 |
| prime radical | C24 | C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
363 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 363 conjugacy classes of subgroups — too many to draw. Only its 199 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,7,9,8,10,11,12);(1,2)(3,4)(13,14,15,17,16,18,19,20);(2,4)160 conjugacy classes — view the full character table.
| maximal subgroup of | #93278, #93281, #93365, … |
| maximal normal subgroup of | #93278, #93281, #93365, … |
| maximal quotient of | C8×#323, C8×#324, C8×D4⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1131);