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) = C23×C4 |
| direct product | C5×C4⋊#338 |
| unambiguous semidirect product | C4⋊(C5×#338) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×C42 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | C2×#95 |
| prime radical | C24×C5 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ G | |
762 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)(7,9)(10,11,13,17,15,18,14,23)(12,19,21,16,22,24,20,25);(10,12,14,21,15,22,13,20)(11,16,17,19,18,25,23,24);(6,7,8,9)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3826);