Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#1896 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C2×C22⋊C4 |
| prime radical | C2×#525 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1257 conjugacy classes of subgroups; only the 263 normal subgroups are shown.
This group has 1257 conjugacy classes of subgroups — too many to draw. Only its 263 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 14 points:
(1,2);(3,4)(5,6)(7,8,10,13,11,14,9,12);(8,12,14,13);(4,6)88 conjugacy classes — view the full character table.
| maximal subgroup of | #108074, #108086, #110440, … |
| maximal normal subgroup of | #108074, #108086, #110440, … |
| maximal quotient of | C2×#9071, C2×#9072, #195806, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13965);