Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
| direct product | C22×#555 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator, Frattini, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
4355 conjugacy classes of subgroups; only the 2967 normal subgroups are shown.
This group has 4355 conjugacy classes of subgroups — too many to draw. Only its 2967 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(3,4);(1,2)(3,4);(5,6)(7,11)(8,12)(9,10)(14,15)(18,19);(6,10)(8,11)(13,14,17,15)(16,19,20,18);(5,7,9,12)(6,11,10,8)(13,15,17,14)(16,19,20,18);(5,8,9,11)(6,12,10,7)(13,16,17,20)(14,18,15,19)100 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#555 |
| maximal subgroup of | C2×#8360, #185477, C2×#8397, … |
| maximal normal subgroup of | C2×#8360, #185477, C2×#8397, … |
| maximal quotient of | C22×#2354, C22×#2346, C22×#2363, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55695);