Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32706 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#32834 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C43×C5 ◁ G | |
720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,9,11)(8,10,12,13)(22,23)(24,28)(25,29)(26,27)(30,36)(31,34)(32,35)(33,37);(6,8)(7,10)(9,12)(11,13)(14,15)(16,18)(17,20)(19,21)(23,27)(25,32)(28,34)(30,33);(7,11)(10,13)(15,18)(16,21)(19,20)(22,24,26,31)(23,28,27,34)(25,33,32,30)(29,37,35,36);(8,12)(10,13)(22,25,26,32)(23,29,27,35)(24,30,31,33)(28,36,34,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 25822);