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×#11374 |
| 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 | |
598 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)(6,7)(8,9)(10,11)(12,21,25,37)(13,26)(14,19)(15,20)(16,30)(17,23,33,24)(18,29)(22,38,39,34)(27,41,40,32)(28,36)(31,35);(7,9)(10,12,16,23,15,25,14,24)(11,17,30,21,20,33,19,37)(13,27,31,39,29,40,28,22)(18,34,36,32,26,38,35,41);(7,9)(10,13,16,31,15,29,14,28)(11,18,30,36,20,26,19,35)(12,22,23,27,25,39,24,40)(17,32,21,38,33,41,37,34)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4362);