Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#21547 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×C4×D4 |
| prime radical | C5×#3050 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C5×M16 ◁ G | |
906 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)(10,11,15,19,16,20,14,30)(12,21,24,32,25,37,23,17)(13,18,29,35,27,34,28,41)(22,36,39,33,26,40,38,31);(10,12,14,23,16,25,15,24)(11,17,19,21,20,32,30,37)(13,26,28,39,27,22,29,38)(18,33,35,40,34,31,41,36);(7,9)(13,27)(18,34)(22,26)(28,29)(31,33)(35,41)(36,40)(38,39);(10,13,15,29,16,27,14,28)(11,18,19,35,20,34,30,41)(12,22,24,39,25,26,23,38)(17,31,21,36,32,33,37,40)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14535);