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×#35340 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C5 |
| prime radical | C2×C5×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
1630 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)(7,11)(8,13)(15,19)(20,21)(25,32)(29,34)(31,36)(35,37);(6,7,10,11)(8,12,13,9)(14,15)(16,20)(17,21)(18,19)(23,27)(24,30)(29,34)(31,36);(6,8)(7,12)(9,11)(10,13)(14,16,18,17)(15,20,19,21)(22,23,26,27)(24,28,30,33)(25,29,32,34)(31,35,36,37);(6,9,10,12)(7,8,11,13)(14,17,18,16)(15,21,19,20)(22,24)(23,28)(25,31)(26,30)(27,33)(29,35)(32,36)(34,37);(22,25,26,32)(23,29,27,34)(24,31,30,36)(28,35,33,37)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 28328);