Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
| direct product | C8×C4⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C23 |
| commutator | C22 | C23×C8 |
| Frattini, cosocle | C22×C4 | C24 |
| socle, prime radical | C23 | C23×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
443 conjugacy classes of subgroups; only the 263 normal subgroups are shown.
This group has 443 conjugacy classes of subgroups — too many to draw. Only its 263 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:
(13,14,15,17,16,18,19,20);(2,4)(5,6,8,10)(7,11,12,9);(5,7,8,12)(6,9,10,11);(1,2,3,4)112 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#333, C8×#332, #96854, … |
| maximal normal subgroup of | C8×#333, C8×#332, #96854, … |
| maximal quotient of | C8×#498, C8×#500, C8×#499, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14699);