Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#20620 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#120 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#379 |
| prime radical | C2×C5×#585 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
3648 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)(6,7,11,16,12,15,10,23)(8,17,19,13,20,28,24,27)(9,14,21,31,22,29,25,30)(18,33,36,37,35,32,34,26);(6,8)(7,13)(9,18)(10,24)(11,19)(12,20)(14,26)(15,27)(16,28)(17,23)(21,36)(22,35)(25,34)(29,37)(30,32)(31,33);(6,9)(7,14)(8,18)(10,25)(11,21)(12,22)(13,26)(15,29)(16,31)(17,32)(19,36)(20,35)(23,30)(24,34)(27,37)(28,33);(7,15)(13,27)(14,29)(16,23)(17,28)(26,37)(30,31)(32,33)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13608);