Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#36732 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C5 |
| prime radical | C25×C5 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
1770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2,3,4,5)(8,12)(11,13)(16,20)(17,21)(22,23,27,28)(24,31,34,29)(25,35,30,26)(32,33,36,37);(7,10)(11,13)(14,15,18,19)(16,21,20,17)(23,28)(25,30)(29,31)(32,36);(6,7)(8,11)(9,10)(12,13)(14,16)(15,17)(18,20)(19,21)(22,24,27,34)(23,29,28,31)(25,32,30,36)(26,33,35,37);(6,7)(8,11)(9,10)(12,13)(14,15)(16,17)(18,19)(20,21)(22,25,27,30)(23,26,28,35)(24,32,34,36)(29,33,31,37);(6,8)(7,11)(9,12)(10,13)(14,17)(15,16)(18,21)(19,20)(22,26)(23,30)(24,33)(25,28)(27,35)(29,36)(31,32)(34,37)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 29720);