Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#12658 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×C42 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C2×#97 |
| prime radical | C23×C5 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ G | |
350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5)(6,7,8,9)(11,16,18,24)(12,20)(14,23)(17,25,19,21);(7,9)(10,11,14,19,13,16,20,25,15,18,23,17,22,24,12,21);(10,12,13,14,15,20,22,23)(11,17,24,25,18,19,16,21)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5646);