Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C22×#501 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1071 conjugacy classes of subgroups; only the 607 normal subgroups are shown.
This group has 1071 conjugacy classes of subgroups — too many to draw. Only its 607 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) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2);(3,4);(5,6,8,10)(7,11,12,9)(14,17)(15,19)(18,20);(5,7,8,12)(6,9,10,11)(14,18)(17,20);(13,14,15,17,16,18,19,20)64 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#501 |
| maximal subgroup of | #112674, #112676, #112856, … |
| maximal normal subgroup of | #112674, #112676, #112856, … |
| maximal quotient of | C22×#1847, C22×#1943, C22×#1944, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53644);