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×#36736 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C5 |
| prime radical | C2×C5×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
1782 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)(22,23,27,24)(25,30,33,28)(26,29,34,31)(32,36,37,35);(7,10)(11,13)(14,15,18,19)(16,21,20,17)(23,24)(28,30)(29,31)(35,36);(6,7)(8,11)(9,10)(12,13)(14,16)(15,17)(18,20)(19,21)(22,24,27,23)(25,30,33,28)(26,31,34,29)(32,36,37,35);(6,7)(8,11)(9,10)(12,13)(14,15)(16,17)(18,19)(20,21)(22,25)(23,28)(24,30)(26,32)(27,33)(29,35)(31,36)(34,37);(6,8)(7,11)(9,12)(10,13)(14,17)(15,16)(18,21)(19,20)(22,26,27,34)(23,29,24,31)(25,32,33,37)(28,35,30,36)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 29724);