Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17411 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#17430 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C23×C5 | C23×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C4×C5×Q8 ◁ G | |
884 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,9,13,10,12,18,20)(8,14,15,21,16,11,17,19)(22,23,26,30,27,31,25,29)(24,28,32,35,33,36,34,37);(6,8,10,16)(7,11,12,14)(9,17,18,15)(13,21,20,19);(22,24,27,33)(23,28,31,36)(25,34,26,32)(29,37,30,35);(22,25,27,26)(23,29,31,30)(24,32,33,34)(28,35,36,37)275 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10418);