Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#1781 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C24 | C24 |
| prime radical | C24×D4 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3127 conjugacy classes of subgroups; only the 327 normal subgroups are shown.
This group has 3127 conjugacy classes of subgroups — too many to draw. Only its 327 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 14 points:
(1,2);(3,4)(5,6)(7,8)(9,13,14,11)(10,12);(7,9)(8,11)(10,14)(12,13);(4,6)64 conjugacy classes — view the full character table.
| maximal subgroup of | #105808, #105810, #106132, … |
| maximal normal subgroup of | #105808, #105810, #106132, … |
| maximal quotient of | C2×#8298, C2×#8299, C2×#8301, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13570);