Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#1978 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #121 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D8 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
847 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 847 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);(4,6)(7,8,12,16,13,15,11,9)(10,18,20,22,21,17,19,14);(3,4,5,6)(7,9,12,15,13,16,11,8)(10,17,20,22,21,18,19,14);(7,10,12,20,13,21,11,19)(8,14,9,17,15,22,16,18)64 conjugacy classes — view the full character table.
| maximal subgroup of | #147415, #147421, #147436, … |
| maximal normal subgroup of | #147415, #147421, #147436, … |
| maximal quotient of | C2×#9668, C2×#9670, #204440, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14234);