Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#21075 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #121 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×#121 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
1014 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)(6,7,9,8)(10,11)(12,21)(13,26)(14,19)(15,30)(16,20)(17,24)(18,29)(22,36)(23,32)(25,35)(27,41)(28,33)(31,37)(34,39)(38,40);(6,8,9,7)(10,12,15,24,16,25,14,23)(11,17,19,21,20,32,30,35)(13,22,28,38,29,39,27,37)(18,31,33,34,26,40,41,36);(10,13,15,28,16,29,14,27)(11,18,19,33,20,26,30,41)(12,22,24,38,25,39,23,37)(17,31,21,34,32,40,35,36);(10,14,16,15)(11,19,20,30)(12,23,25,24)(13,27,29,28)(17,21,32,35)(18,33,26,41)(22,37,39,38)(31,34,40,36)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14063);