Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#10872 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #121 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #380 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
730 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)(10,11,15,19,16,20,14,30)(12,21,23,17,25,32,24,33)(13,26,28,34,29,40,27,18)(22,31,36,39,38,41,37,35);(6,7,8,9)(10,12)(11,17)(13,22)(14,24)(15,23)(16,25)(18,31)(19,32)(20,33)(21,30)(26,39)(27,37)(28,36)(29,38)(34,41)(35,40);(10,13,14,27,16,29,15,28)(11,18,19,26,20,34,30,40)(12,22,24,37,25,38,23,36)(17,31,32,39,33,41,21,35)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3860);