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×#21545 |
| 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)(10,11,15,19,16,20,14,29)(12,21,24,31,25,35,23,17)(13,18,28,33,26,32,27,39)(22,34,38,40,36,41,37,30);(10,12,14,23,16,25,15,24)(11,17,19,21,20,31,29,35)(13,22,27,37,26,36,28,38)(18,30,33,34,32,40,39,41);(7,9)(13,26)(18,32)(22,36)(27,28)(30,40)(33,39)(34,41)(37,38);(10,13,15,28,16,26,14,27)(11,18,19,33,20,32,29,39)(12,22,24,38,25,36,23,37)(17,30,21,34,31,40,35,41)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14533);