Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18562 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18601 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C22×C5×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
910 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)(6,7,11,14)(8,15,18,21)(9,13,10,19)(12,16,20,17)(22,23,27,31)(24,32,33,36)(25,30,26,29)(28,35,37,34);(6,8,11,18)(7,12,14,20)(9,17,10,16)(13,21,19,15)(22,24,27,33)(23,28,31,37)(25,35,26,34)(29,36,30,32);(23,29)(25,26)(28,36)(30,31)(32,37)(34,35);(6,9,11,10)(7,13,14,19)(8,16,18,17)(12,15,20,21)(24,33)(28,37)(32,36)(34,35)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11589);