Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×C4×#99 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22⋊C4 |
| commutator | C22 | C22×C42 |
| Frattini, cosocle | C24 | C24 |
| socle | C23 | C2×C22⋊C4 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
1121 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 14 points:
(1,2);(3,4,5,6);(7,8)(9,13,14,11)(10,12);(7,9)(8,11)(10,14)(12,13)88 conjugacy classes — view the full character table.
| maximal subgroup of | #105574, #105575, #105610, … |
| maximal normal subgroup of | #105574, #105575, #105610, … |
| maximal quotient of | C2×C4×#323, C2×C4×#324, C4×#1459, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13378);