Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17386 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25×C4 |
| direct product | C5×#21220 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×#117 ◁ G | |
1454 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)(7,9)(10,11,14,19,15,18,20,32)(12,16,23,28,22,29,26,33)(13,17,25,31,24,30,34,40)(21,27,36,39,35,38,37,41);(10,12,20,23,15,22,14,26)(11,16,19,33,18,29,32,28)(13,21,34,36,24,35,25,37)(17,27,31,41,30,38,40,39);(6,7,8,9)(10,13)(11,17)(12,21)(14,25)(15,24)(16,27)(18,30)(19,31)(20,34)(22,35)(23,36)(26,37)(28,39)(29,38)(32,40)(33,41);(13,24)(17,30)(21,35)(25,34)(27,38)(31,40)(36,37)(39,41)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14208);