Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C42 |
| direct product | C4×#353 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C22×D4 | C23 |
| socle | C22 | C2×#99 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
360 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2,3,4);(5,6)(7,10,12,9)(8,11);(6,9)(7,12)(10,11)52 conjugacy classes — view the full character table.
| maximal subgroup of | #94143, #94147, #94153, … |
| maximal normal subgroup of | #94143, #94147, #94153, … |
| maximal quotient of | C4×#1318, C4×#1325, #128368, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5613);