Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#1811 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C2×C4⋊C4 |
| prime radical | C25 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
719 conjugacy classes of subgroups; only the 271 normal subgroups are shown.
This group has 719 conjugacy classes of subgroups — too many to draw. Only its 271 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4)(5,6)(7,8,11,14,12,15,10,20)(9,16,19,21,17,22,18,13);(7,9,10,18,12,17,11,19)(8,13,14,16,15,21,20,22);(4,6)(9,17)(13,21)(16,22)(18,19)64 conjugacy classes — view the full character table.
| maximal subgroup of | #116449, #116464, #117017, … |
| maximal normal subgroup of | #116449, #116464, #117017, … |
| maximal quotient of | C2×#8597, #188710, C2×#8598, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13654);