Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24 |
| direct product | C2×#2183 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C23 |
| commutator | C22 | C23×C8 |
| Frattini, cosocle | C2×C8 | C24 |
| socle, prime radical | C23 | C22×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
271 conjugacy classes of subgroups; only the 215 normal subgroups are shown.
This group has 271 conjugacy classes of subgroups — too many to draw. Only its 215 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) = 256 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4)(5,10)(6,8)(7,9)(11,12,14,18,15,13,20,21,16,19,23,26,24,22,17,25);(3,5,7,6)(4,8,9,10);(3,6,7,5)(4,8,9,10)(11,13,14,21,15,19,20,26,16,22,23,25,24,12,17,18)112 conjugacy classes — view the full character table.
| maximal subgroup of | #128509, #128515, #128678, … |
| maximal normal subgroup of | #128509, #128515, #128678, … |
| maximal quotient of | C2×#12663, C2×#12662, C4×#2183, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26461);