Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#12671 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×D8 |
| commutator | C8 | C2×C42×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C4×D8 |
| prime radical | C23×C5 | #135 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
310 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)(10,11,15,20,16,21,14,19,17,22,31,38,30,37,13,29)(12,23,26,18,27,34,32,35,28,39,24,36,33,41,25,40);(6,7,8,9)(10,12)(11,18)(13,24)(14,26)(15,32)(16,33)(17,28)(19,34)(20,39)(21,40)(22,36)(23,38)(25,31)(27,30)(29,41)(35,37);(10,13,30,31,17,14,16,15)(11,19,37,20,22,29,21,38)(12,24,27,25,28,26,33,32)(18,34,35,39,36,41,40,23)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5659);