Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19014 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#19049 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3561 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
886 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)(6,7,12,17)(8,18,9,20)(10,15,11,16)(13,21,14,19)(22,23,27,25)(24,30,32,37)(26,34,36,29)(28,35,31,33);(6,8,12,9)(7,13,17,14)(10,21,11,19)(15,18,16,20)(22,24,27,32)(23,28,25,31)(26,35,36,33)(29,37,34,30);(6,9,12,8)(7,14,17,13)(10,21,11,19)(15,18,16,20)(22,25,27,23)(24,31,32,28)(26,29,36,34)(30,35,37,33);(6,10,12,11)(7,15,17,16)(8,19,9,21)(13,20,14,18)(24,32)(28,31)(30,37)(33,35)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12037);