Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | Aut(G) = C24×D4×#120 (order 4096) · Inn(G) = C22×D4 · Out(G) = C24×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | D4×#104 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | D4×#41 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
804 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 12 points:
(1,2)(3,4);(2,4);(5,6,8,11,9,7,12,10);(5,7)(6,9)(8,10)(11,12)70 conjugacy classes — view the full character table.
| maximal subgroup of | #221064, #221065, #221070, … |
| maximal normal subgroup of | #221064, #221065, #221070, … |
| maximal quotient of | D4×D4⋊C8, #219797, D4×Q8⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15421);