Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11336 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #127 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #399 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
694 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)(10,11,15,19,16,21,14,20)(12,22,26,32,25,33,24,17)(13,18,28,36,30,37,29,35)(23,34,39,31,40,38,27,41);(6,7)(8,9)(10,12,14,24,16,25,15,26)(11,17,19,22,21,32,20,33)(13,27,28,23,30,39,29,40)(18,34,35,41,37,38,36,31);(7,9)(10,13)(11,18)(12,23,25,40)(14,29)(15,28)(16,30)(17,31,32,41)(19,36)(20,35)(21,37)(22,38,33,34)(24,27,26,39)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4324);