Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×C4×#107 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | D4 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle, prime radical | C24 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
679 conjugacy classes of subgroups; only the 431 normal subgroups are shown.
This group has 679 conjugacy classes of subgroups — too many to draw. Only its 431 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4,5,6);(7,8,9,10)(12,15)(13,16)(17,18);(11,12,16,18,14,17,13,15)112 conjugacy classes — view the full character table.
| maximal subgroup of | #108852, #108853, #108856, … |
| maximal normal subgroup of | #108852, #108853, #108856, … |
| maximal quotient of | C2×C4×#335, C2×#8278, C2×#8274, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13436);