Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5×C8 · Schur multiplier (H2) = C25 |
| direct product | C5×#33401 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | C22×D4 |
| commutator | C4 | C23×C5×C8 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C2×C5 | C22×C4⋊C8 |
| prime radical | C2×C5×#41 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C2×C5×#131 ◁ G | |
684 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)(6,7,11,17,12,16,22,32,13,18,21,14,10,27,8,19)(9,15,25,31,26,30,34,37,23,29,35,28,24,36,20,33);(6,8,10,21,13,22,12,11)(7,14,16,19,18,17,27,32)(9,20,24,35,23,34,26,25)(15,28,30,33,29,31,36,37);(6,9)(7,15)(8,20)(10,24)(11,25)(12,26)(13,23)(14,28)(16,30)(17,31)(18,29)(19,33)(21,35)(22,34)(27,36)(32,37);(9,23)(15,29)(20,34)(24,26)(25,35)(28,31)(30,36)(33,37)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26389);