Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C5 · Schur multiplier (H2) = C213 |
| direct product | C5×#62761 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C26×C5 |
| socle | C22×C5 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ G | |
6518 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)(8,14)(9,13)(10,12)(11,15)(16,20)(17,19)(18,21)(22,23,27,31)(24,28,33,36)(25,30,26,29)(32,37,34,35);(6,8)(7,11)(9,16)(10,15)(12,14)(13,18)(17,21)(19,20)(22,24)(23,28)(25,32)(26,34)(27,33)(29,35)(30,37)(31,36);(8,15)(11,14)(16,21)(18,20)(22,25,27,26)(23,29,31,30)(24,32,33,34)(28,35,36,37);(7,12)(11,14)(13,19)(18,20)(24,33)(28,36)(32,34)(35,37);(22,26,27,25)(23,30,31,29)(24,34,33,32)(28,37,36,35);(6,9,10,17)(7,13,12,19)(8,16,15,21)(11,18,14,20)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55749);