Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#23652 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #397 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#397 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
888 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,11,16,12,17,10,15)(8,18,20,13,22,29,21,30)(9,23,25,31,26,32,27,14)(19,34,36,37,24,28,35,33);(6,8)(7,13)(9,24)(10,20)(11,21)(12,22)(14,28)(15,29)(16,18)(17,30)(19,26)(23,37)(25,36)(27,35)(31,34)(32,33);(6,9,10,25,12,26,11,27)(7,14,15,23,17,31,16,32)(8,19,21,36,22,24,20,35)(13,28,18,33,30,34,29,37);(6,10,12,11)(7,15,17,16)(8,20,22,21)(9,25,26,27)(13,29,30,18)(14,23,31,32)(19,35,24,36)(28,37,34,33)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16640);