Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#24462 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#578 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
1302 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)(10,11,16,20)(12,22,25,32)(13,26,30,18)(14,24,17,33)(15,21,34,19)(23,39,41,36)(27,35,40,28)(29,37,31,38);(10,12,16,25)(11,17,20,14)(13,27,30,40)(15,33,34,24)(18,36,26,39)(19,32,21,22)(23,29,41,31)(28,37,35,38);(7,9)(10,13,16,30)(11,18,20,26)(12,23,25,41)(14,28,17,35)(15,29,34,31)(19,37,21,38)(22,36,32,39)(24,40,33,27);(10,14)(11,12)(13,28)(15,22)(16,17)(18,23)(19,33)(20,25)(21,24)(26,41)(27,37)(29,36)(30,35)(31,39)(32,34)(38,40)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17450);