Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24 |
| direct product | C8×#126 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C23 |
| commutator | C22 | C23×C8 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C23×C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
393 conjugacy classes of subgroups; only the 215 normal subgroups are shown.
This group has 393 conjugacy classes of subgroups — too many to draw. Only its 215 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 — 4 generators on 24 points:
(17,18,19,21,20,22,23,24);(1,2)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16);(1,4,5,3)(2,6,7,8)(10,14)(13,16)112 conjugacy classes — view the full character table.
| Fitting subgroup of | C8×#126⋊C3 |
| maximal subgroup of | C16×#126, #496070, #496073, … |
| maximal normal subgroup of | C16×#126, #496070, #496073, … |
| maximal quotient of | C8×#388, C8×#387, C8×#383, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14652);