Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C4×#453 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×#120 |
| prime radical | C2×#453 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
789 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 789 conjugacy classes of subgroups — too many to draw. Only its 189 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 12 points:
(1,2,3,4);(6,9)(7,11)(10,12);(5,6)(7,12)(8,10)(9,11);(7,11)(9,12)64 conjugacy classes — view the full character table.
| maximal subgroup of | #136821, #136823, #136849, … |
| maximal normal subgroup of | #136821, #136823, #136849, … |
| maximal quotient of | C8×#453, #208748, #208756, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14457);