Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18284 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18295 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
940 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,10,13)(8,14,16,21)(9,17,19,12)(11,18,20,15)(22,23)(24,27)(25,28)(26,29)(30,36)(31,34)(32,35)(33,37);(6,8,10,16)(7,11,13,20)(9,18,19,15)(12,21,17,14)(22,24)(23,27)(25,30)(26,31)(28,33)(29,34)(32,37)(35,36);(22,25,26,32)(23,28,29,35)(24,30,31,37)(27,33,34,36);(23,29)(27,34)(28,35)(33,36)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11283);