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) = C26 |
| direct product | C5×#10760 |
| 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 | C22×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C5×M16 ◁ G | |
1170 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)(10,11,14,21,16,20,15,19)(12,22,24,17,26,33,25,32)(13,27,29,35,30,18,28,34)(23,37,39,31,40,36,38,41);(7,9)(10,12,15,25,16,26,14,24)(11,17,21,33,20,32,19,22)(13,23,29,39,30,40,28,38)(18,31,35,37,27,41,34,36);(10,13)(11,18)(12,23)(14,28)(15,29)(16,30)(17,31)(19,34)(20,27)(21,35)(22,36)(24,38)(25,39)(26,40)(32,41)(33,37)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3748);