Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C22×Q8⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | D4 |
| commutator | C4 | C23×C8 |
| Frattini, cosocle | C42 | C24 |
| socle, prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
727 conjugacy classes of subgroups; only the 351 normal subgroups are shown.
This group has 727 conjugacy classes of subgroups — too many to draw. Only its 351 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) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2);(3,4);(6,9)(7,11)(10,12)(13,14,15,17,16,18,19,20);(5,6,8,10)(7,12,11,9)112 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×Q8⋊C8 |
| maximal subgroup of | #105669, #105671, #105675, … |
| maximal normal subgroup of | #105669, #105671, #105675, … |
| maximal quotient of | C22×#1271, C22×#1272, C2×#7562, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10420);