Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#35321 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C5 |
| prime radical | C24×C5 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
1558 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)(7,8)(11,12)(15,18)(19,21)(22,23,27,25)(24,28,30,31)(26,33,34,29)(32,37,36,35);(6,7,10,8)(9,11,13,12)(14,15)(16,19)(17,18)(20,21)(23,25)(24,30)(29,33)(32,36);(6,8,10,7)(9,12,13,11)(22,24)(23,28)(25,31)(26,32)(27,30)(29,35)(33,37)(34,36);(6,9,10,13)(7,11,8,12)(14,16,17,20)(15,19,18,21)(22,25,27,23)(24,31,30,28)(26,33,34,29)(32,37,36,35);(22,26,27,34)(23,29,25,33)(24,32,30,36)(28,35,31,37)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 28309);