Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | C22×#537 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator, Frattini, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
6135 conjugacy classes of subgroups; only the 3183 normal subgroups are shown.
This group has 6135 conjugacy classes of subgroups — too many to draw. Only its 3183 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 16 points:
(3,4);(1,2)(3,4);(6,8)(9,10)(11,15)(12,13)(14,16);(10,13)(14,15);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16);(9,11,12,16)(10,14,13,15)88 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#537 |
| maximal subgroup of | C2×#8404, #185933, #185972, … |
| maximal normal subgroup of | C2×#8404, #185933, #185972, … |
| maximal quotient of | C22×#2316, C22×#2311, C22×#2325, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55663);