Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#20351 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C4 | C22×C42×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C22×C4⋊C4 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C5×#41 ◁ G | |
1978 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,14,20,16,21,15,19)(12,22,24,17,26,31,25,32)(13,18,28,35,27,33,29,34)(23,36,38,30,37,41,39,40);(7,9)(10,12,15,25,16,26,14,24)(11,17,20,31,21,32,19,22)(13,23,29,39,27,37,28,38)(18,30,35,41,33,40,34,36);(10,13)(11,18)(12,23)(14,28)(15,29)(16,27)(17,30)(19,34)(20,35)(21,33)(22,36)(24,38)(25,39)(26,37)(31,41)(32,40);(13,27)(18,33)(23,37)(28,29)(30,40)(34,35)(36,41)(38,39)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13339);