Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#12586 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#99 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C22×Q8 | C23×C5 |
| prime radical | C24×C5 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C24×C5 ◁ C23×C5×Q8 ◁ G | |
746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(10,11,14,18,15,19,29,35)(12,20,23,37,21,36,27,40)(13,24,25,34,26,17,28,33)(16,30,32,38,31,39,41,22);(6,7,8,9)(10,12)(11,16,19,31)(13,22)(14,27)(15,21)(17,20,24,36)(18,32,35,41)(23,29)(25,39)(26,38)(28,30)(33,37,34,40);(12,21)(16,31)(20,36)(22,38)(23,27)(30,39)(32,41)(37,40)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5574);