Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#60344 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C25 |
| commutator | C22 | C24×C4×C5 |
| Frattini, cosocle | C2×C4 | C25×C5 |
| socle | C22×C5 | C24×C4 |
| prime radical | C5×#537 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ G | |
1664 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2,3,4,5)(6,7)(8,14)(9,12)(10,13)(11,16)(15,20)(17,19)(18,21)(22,23,28,30,27,31,25,35)(24,32,33,37,34,29,36,26);(6,8,10,16)(7,11,13,14)(9,17,18,15)(12,20,21,19)(22,24,27,34)(23,29,31,32)(25,36,28,33)(26,35,37,30);(6,9,10,18)(7,12,13,21)(8,15,16,17)(11,19,14,20)(22,25,27,28)(23,30,31,35)(24,33,34,36)(26,29,37,32);(7,13)(11,14)(12,21)(19,20)(23,31)(24,34)(30,35)(33,36);(22,26,28,32,27,37,25,29)(23,24,30,33,31,34,35,36)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53332);