Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#17621 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×D4 | D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C4×C5×D4 ◁ G | |
990 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)(7,12,14,27)(8,17)(9,21)(13,29,30,35)(15,28,16,22)(18,20)(23,25)(26,32,37,31);(6,7,10,15)(8,13,17,31)(9,22,21,27)(11,16,24,14)(12,25,28,23)(18,30,20,32)(19,35,33,37)(26,34,29,36);(6,8)(7,13)(9,19)(10,17)(11,20)(12,26)(14,30)(15,31)(16,32)(18,24)(21,33)(22,35)(23,36)(25,34)(27,37)(28,29);(8,18)(13,30)(17,20)(19,34)(26,37)(29,35)(31,32)(33,36)245 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10609);