Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23912 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#24229 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3497 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
1330 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)(7,8)(10,11)(12,16)(13,23)(14,26)(15,19)(17,25)(18,29)(20,37)(21,35)(22,32)(24,36)(27,31)(28,34)(30,39)(33,40)(38,41);(6,7,9,8)(10,12,15,22)(11,16,19,32)(13,20,24,38)(14,27,29,21)(17,33,34,30)(18,35,26,31)(23,37,36,41)(25,40,28,39);(10,13,15,24)(11,17,19,34)(12,20,22,38)(14,28,29,25)(16,30,32,33)(18,36,26,23)(21,40,27,39)(31,41,35,37);(6,8,9,7)(13,24)(17,34)(20,38)(23,36)(25,28)(30,33)(37,41)(39,40)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17217);