Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C22×#331 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| prime radical | C22×#127 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1011 conjugacy classes of subgroups; only the 311 normal subgroups are shown.
This group has 1011 conjugacy classes of subgroups — too many to draw. Only its 311 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 — 4 generators on 20 points:
(1,2);(3,4);(5,6,7,10,8,9,11,12)(13,14)(15,17)(16,19)(18,20);(6,9)(7,11)(14,17)(15,20)(18,19)76 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#331 |
| maximal subgroup of | #106032, #108334, #110375, … |
| maximal normal subgroup of | #106032, #108334, #110375, … |
| maximal quotient of | C22×#1270, C2×#7569, C2×#7640, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10681);