Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#2139 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| prime radical | C22×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
607 conjugacy classes of subgroups; only the 231 normal subgroups are shown.
This group has 607 conjugacy classes of subgroups — too many to draw. Only its 231 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) = 128 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4,5,6)(8,12)(9,15)(10,17)(13,20)(14,21)(16,19)(18,22);(7,8)(9,16)(10,19)(11,13)(12,15)(14,17)(18,21)(20,22);(8,13)(12,20)(14,19)(16,21)64 conjugacy classes — view the full character table.
| maximal subgroup of | #146748, #146764, #146774, … |
| maximal normal subgroup of | #146748, #146764, #146774, … |
| maximal quotient of | C2×#11592, C2×#11660, #481267, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26367);