Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) = C2×#120×#525 (order 4096) · Inn(G) = C24 · Out(G) = C22×D42 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
| direct product | D4×M32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C24 |
| commutator, socle | C22 | C23×C8 |
| prime radical | C22×D4 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
367 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4);(2,4);(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20);(6,14)(8,16)(10,18)(12,20)100 conjugacy classes — view the full character table.
| maximal subgroup of | #128702, #128717, #134944, … |
| maximal normal subgroup of | #128702, #128717, #134944, … |
| maximal quotient of | D8×M32, SD16×M32, Q16×M32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26425);