Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18992 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#19012 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
844 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)(7,12)(9,10)(13,20)(14,17)(15,18)(19,21)(22,23)(24,31)(25,29)(26,34)(27,30)(28,32)(33,36)(35,37);(6,7,11,14)(8,13,16,21)(9,12,10,17)(15,19,18,20)(22,23)(24,28)(25,29)(26,34)(27,30)(31,35)(32,36)(33,37);(6,8)(7,13)(9,15)(10,18)(11,16)(12,19)(14,21)(17,20)(22,24)(23,28)(25,32)(26,35)(27,33)(29,36)(30,37)(31,34);(6,9,11,10)(7,12,14,17)(8,15,16,18)(13,19,21,20)(22,25,27,26)(23,29,30,34)(24,32,33,35)(28,36,37,31)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12000);