Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#25194 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3013 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
980 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)(10,11,16,20,17,22,15,21)(12,23,25,18,13,27,26,34)(14,29,30,19,32,36,31,37)(24,33,40,41,28,35,39,38);(6,7)(8,9)(10,12,16,25,17,13,15,26)(11,18,20,27,22,34,21,23)(14,28,30,39,32,24,31,40)(19,35,36,38,37,33,29,41);(7,9)(10,13,17,12)(11,18,22,34)(14,28,32,24)(15,26,16,25)(19,33,37,35)(20,23,21,27)(29,41,36,38)(30,40,31,39);(10,14)(11,19)(12,24)(13,28)(15,31)(16,30)(17,32)(18,33)(20,36)(21,29)(22,37)(23,38)(25,40)(26,39)(27,41)(34,35)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 18182);