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×#19036 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3559 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
880 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)(6,7)(8,16)(9,13)(10,14)(11,15)(12,17)(18,20)(19,21)(22,23,27,30)(24,31,34,37)(25,29,26,35)(28,32,36,33);(6,8)(7,12)(9,19)(10,17)(11,18)(13,16)(14,20)(15,21)(22,24,27,34)(23,28,30,36)(25,33,26,32)(29,37,35,31);(6,8)(7,12)(9,17)(10,19)(11,18)(13,20)(14,16)(15,21)(22,23)(24,28)(25,29)(26,35)(27,30)(31,32)(33,37)(34,36);(6,9,11,10)(7,13,15,14)(8,17,18,19)(12,20,21,16)(22,25,27,26)(23,29,30,35)(24,32,34,33)(28,31,36,37)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12024);