Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#1836 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
527 conjugacy classes of subgroups; only the 271 normal subgroups are shown.
This group has 527 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) = 256 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4,5,6)(7,8)(9,12)(10,11)(13,14)(15,16,20,17)(18,19,22,21);(8,11)(12,14)(15,17,19,22,20,16,21,18);(7,9,10,13)(8,12,11,14)(15,18,19,17,20,22,21,16)64 conjugacy classes — view the full character table.
| maximal subgroup of | #110000, #117825, #117826, … |
| maximal normal subgroup of | #110000, #117825, #117826, … |
| maximal quotient of | C2×#8786, C2×#8789, #191181, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13749);