Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19254 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#19277 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C2×C5×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
638 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)(6,7,10,13)(8,14,17,21)(9,18,15,12)(11,19,20,16)(22,23)(24,32)(25,31)(26,28)(27,30)(29,33)(34,37)(35,36);(6,8,15,19,10,17,9,16)(7,11,18,21,13,20,12,14)(23,28)(29,36)(30,31)(32,37);(22,24)(23,29)(25,33)(26,34)(27,35)(28,36)(30,32)(31,37);(22,25,26,27)(23,30,28,31)(24,33,34,35)(29,32,36,37)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12265);