Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19755 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#19863 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3303 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C42×C5 ◁ G | |
594 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(7,11)(9,17)(12,18)(13,14)(15,21)(19,20)(22,23,27,30)(24,28,26,34)(25,32,33,36)(29,35,37,31);(6,7,10,14)(8,12,16,20)(9,13,17,11)(15,19,21,18)(22,24,27,26)(23,28,30,34)(25,31,33,35)(29,36,37,32);(6,8,10,16)(7,12,14,20)(9,15,17,21)(11,18,13,19)(22,25)(23,29)(24,31)(26,35)(27,33)(28,36)(30,37)(32,34);(7,13)(9,17)(11,14)(12,19)(15,21)(18,20)(22,24,27,26)(23,28,30,34)(25,31,33,35)(29,36,37,32)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12851);