Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×C5×#1393 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | #99 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C22×D4 | C23×C5 |
| prime radical | C23×C5×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C24×C5 ◁ C23×C5×D4 ◁ G | |
1154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2)(15,16,17,18,19);(7,8)(9,12,14,11)(10,13);(3,4,5,6)(8,11,13,12)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5568);