Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 3145728 |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C28×C4 |
| direct product | C22×#431 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C42 | C22 |
| commutator | C2 | C24×C8 |
| Frattini, cosocle | C2×C4 | C25 |
| socle | C24 | C22×C4 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1543 conjugacy classes of subgroups.
| 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)(13,14,15,17,16,18,19,20);(6,9)(10,12);(5,7,8,11)(6,10,9,12)160 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#431 |
| maximal subgroup of | #105481, #105490, #105492, … |
| maximal normal subgroup of | #105481, #105490, #105492, … |
| maximal quotient of | C22×#1744, C22×C4×C22⋊C8, C22×C4×C4⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53102);