Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#1897 |
| 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 | |
977 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 977 conjugacy classes of subgroups — too many to draw. Only its 247 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)(8,13,15,21)(10,18)(11,19)(12,17)(14,16,20,22);(4,6)(9,17)(13,21)(16,22)(18,19)64 conjugacy classes — view the full character table.
| maximal subgroup of | #108162, #108175, #112293, … |
| maximal normal subgroup of | #108162, #108175, #112293, … |
| maximal quotient of | C2×#9073, C2×#9074, #195809, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13966);