Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#2980 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| socle | C23 | C23×C4 |
| prime radical | C2×#532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1093 conjugacy classes of subgroups; only the 703 normal subgroups are shown.
This group has 1093 conjugacy classes of subgroups — too many to draw. Only its 703 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12,14,18,15,19,22,26)(13,16,21,23,20,17,24,25);(3,5)(4,7)(6,10)(8,9)(11,13)(12,16)(14,21)(15,20)(17,19)(18,23)(22,24)(25,26);(5,10)(7,9)(11,13)(12,17)(14,21)(15,20)(16,19)(18,25)(22,24)(23,26);(4,8)(7,9)(13,20)(16,17)(21,24)(23,25)76 conjugacy classes — view the full character table.
| maximal subgroup of | #107354, #107359, #107457, … |
| maximal normal subgroup of | #107354, #107359, #107457, … |
| maximal quotient of | C2×#20532, C2×#20539, C2×#20563, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53216);