Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#10961 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×C22⋊C4 |
| commutator | C23 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | C4×C22⋊C4 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C5×C8 ◁ G | |
926 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)(6,7)(8,9)(10,11,14,19,15,18,27,31)(12,16,22,29,21,30,25,34)(13,17,24,32,23,33,26,35)(20,36,37,41,38,28,39,40);(7,9)(10,12)(11,16,18,30)(13,20)(14,25)(15,21)(17,28,33,36)(19,34,31,29)(22,27)(23,38)(24,39)(26,37)(32,41,35,40);(10,13,14,26,15,23,27,24)(11,17,31,32,18,33,19,35)(12,20,25,37,21,38,22,39)(16,28,29,41,30,36,34,40)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3949);