Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C4×#360 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #102 |
| commutator | C8 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×#102 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
233 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 233 conjugacy classes of subgroups — too many to draw. Only its 97 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) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4);(6,10,11,19)(7,13)(8,16)(12,14,17,20)(15,18);(5,6)(7,14)(8,17)(9,11)(10,18)(12,16)(13,19)(15,20)64 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#360, #128691, #128710, … |
| maximal normal subgroup of | C8×#360, #128691, #128710, … |
| maximal quotient of | C4×#1333, C4×#1334, #128590, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5631);