Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 3145728 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C22×#484 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C25 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1331 conjugacy classes of subgroups; only the 567 normal subgroups are shown.
This group has 1331 conjugacy classes of subgroups — too many to draw. Only its 567 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 — 5 generators on 20 points:
(1,2);(3,4);(5,6)(7,10)(8,9)(11,12)(13,14,15,17)(16,18,20,19);(6,9)(10,12)(14,17)(18,19);(5,7,8,11)(6,10,9,12)(14,18)(16,20)(17,19)76 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#484 |
| maximal subgroup of | #114510, #114737, #114789, … |
| maximal normal subgroup of | #114510, #114737, #114789, … |
| maximal quotient of | C22×#1824, C22×#1852, C22×#1872, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53493);