Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C4×C5×#364 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5×C8 | D4 |
| commutator | C4 | C2×C4×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C2×C4⋊C8 |
| prime radical | C23×C5 | C2×M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4)(21,22,23,24,25);(5,6,9,14,10,13,18,20,11,15,17,12,8,19,7,16);(5,7,8,17,11,18,10,9)(6,12,13,16,15,14,19,20)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5643);