Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 3145728 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C22×#451 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C25 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
2491 conjugacy classes of subgroups; only the 647 normal subgroups are shown.
This group has 2491 conjugacy classes of subgroups — too many to draw. Only its 647 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2);(3,4);(5,6)(7,8)(9,10,13,16)(11,17,19,24)(12,20,22,15)(14,21,23,18);(9,11,13,19)(10,14,16,23)(12,21,22,18)(15,24,20,17);(6,8)76 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#451 |
| maximal subgroup of | #114511, #114736, #114788, … |
| maximal normal subgroup of | #114511, #114736, #114788, … |
| maximal quotient of | C22×#1602, C22×#1614, C22×#1605, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53338);