Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62817 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C5 · Schur multiplier (H2) = C213 |
| direct product | C5×#62837 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C5 |
| prime radical | C5×#3444 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
6428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 6 generators on 37 points:
(1,2,3,4,5)(6,7,12,17)(8,15,10,13)(9,19,18,14)(11,16,20,21)(24,30)(27,34)(31,36)(35,37);(6,8,12,10)(7,13,17,15)(9,20,18,11)(14,21,19,16)(22,23,26,29)(24,27,30,34)(25,32,33,28)(31,37,36,35);(6,9)(7,14)(8,11)(10,20)(12,18)(13,16)(15,21)(17,19)(22,24)(23,27)(25,31)(26,30)(28,35)(29,34)(32,37)(33,36);(6,10,12,8)(7,15,17,13)(9,20,18,11)(14,21,19,16);(6,11,12,20)(7,16,17,21)(8,18,10,9)(13,19,15,14)(22,25,26,33)(23,28,29,32)(24,31,30,36)(27,35,34,37);(6,7)(8,13)(9,14)(10,15)(11,16)(12,17)(18,19)(20,21)(22,24)(23,27)(25,31)(26,30)(28,35)(29,34)(32,37)(33,36)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55825);