Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18930 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18977 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3547 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
866 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)(8,17)(9,14)(10,15)(11,20)(12,16)(13,18)(19,21)(22,23,28,31)(24,34,30,37)(25,35,29,36)(26,33,27,32);(6,8,12,9)(7,13,16,21)(10,18,11,19)(14,20,17,15)(22,24,28,30)(23,29,31,25)(26,36,27,35)(32,37,33,34);(6,9,12,8)(7,14,16,17)(10,18,11,19)(13,20,21,15)(22,25,28,29)(23,30,31,24)(26,34,27,37)(32,35,33,36);(6,10,12,11)(7,15,16,20)(8,18,9,19)(13,14,21,17)(23,31)(25,29)(32,33)(34,37)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11965);