Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19755 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#19774 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3294 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C42×C5 ◁ G | |
650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,10,13)(8,14,17,20)(9,18,19,12)(11,15,21,16)(22,23)(24,27)(25,28)(26,30)(29,33)(31,34)(32,36)(35,37);(6,8,10,17)(7,11,13,21)(9,15,19,16)(12,20,18,14)(22,24,26,32)(23,27,30,36)(25,31,33,37)(28,34,29,35);(23,28)(25,33)(27,34)(29,30)(31,37)(35,36);(8,15)(9,19)(11,20)(12,18)(14,21)(16,17)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12762);