Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C42 |
| direct product | C5×#10866 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #117 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #380 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C5×M16 ◁ 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)(12,20,23,16)(13,24)(14,27)(15,18)(17,25)(19,28)(21,34,36,38)(22,30,37,33)(26,39)(29,35,32,41)(31,40);(6,7,8,9)(10,12)(11,16)(13,21)(14,22)(15,23)(17,29)(18,20)(19,30)(24,38)(25,35)(26,36)(27,33)(28,37)(31,32)(34,39)(40,41);(10,13,14,25,15,26,28,40)(11,17,19,24,18,31,27,39)(12,21,22,35,23,36,37,41)(16,29,30,38,20,32,33,34)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3854);