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×#36819 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C5 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
1680 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)(8,12)(11,13)(15,19)(17,21)(22,23,27,30)(24,28,32,26)(25,33,35,29)(31,34,37,36);(7,10)(11,13)(15,19)(16,20)(22,24,27,32)(23,26,30,28)(25,31,35,37)(29,34,33,36);(6,7)(8,11)(9,10)(12,13)(14,15,18,19)(16,17,20,21)(22,24)(23,28)(25,31)(26,30)(27,32)(29,36)(33,34)(35,37);(6,7)(8,11)(9,10)(12,13)(14,16,18,20)(15,17,19,21)(22,25,27,35)(23,29,30,33)(24,31,32,37)(26,34,28,36);(6,8)(7,11)(9,12)(10,13)(14,17)(15,20)(16,19)(18,21)(22,26,27,28)(23,24,30,32)(25,34,35,36)(29,31,33,37)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 29807);