Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C23×C42 |
| direct product | C4×#340 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | D4 |
| commutator | C4 | C43 |
| Frattini, cosocle | C23×C4 | C23 |
| socle, prime radical | C24 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
591 conjugacy classes of subgroups; only the 311 normal subgroups are shown.
This group has 591 conjugacy classes of subgroups — too many to draw. Only its 311 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 — 3 generators on 20 points:
(1,2,3,4);(5,6)(7,8)(9,10,11,12)(14,18)(15,16)(19,20);(6,8)(13,14,16,19,17,20,15,18)112 conjugacy classes — view the full character table.
| maximal subgroup of | #93418, #93420, C8×#340, … |
| maximal normal subgroup of | #93418, #93420, C8×#340, … |
| maximal quotient of | C4×#1277, #105273, #105275, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3690);