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×#35315 |
| 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 | |
1602 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)(6,7,11,10)(8,13,9,12)(14,15,18,21)(16,19,17,20)(22,23,26,29)(24,27,32,30)(25,33,34,28)(31,36,37,35);(6,8,11,9)(7,12,10,13)(14,15)(16,19)(17,20)(18,21)(22,23,26,29)(24,30,32,27)(25,28,34,33)(31,36,37,35);(6,9,11,8)(7,13,10,12)(14,16,18,17)(15,19,21,20)(22,23)(24,27)(25,28)(26,29)(30,32)(31,35)(33,34)(36,37);(6,10,11,7)(8,13,9,12)(14,17,18,16)(15,20,21,19)(22,24)(23,27)(25,31)(26,32)(28,35)(29,30)(33,36)(34,37);(22,25,26,34)(23,28,29,33)(24,31,32,37)(27,35,30,36)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 28303);