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×#19265 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C23×C5 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
606 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)(6,7,11,14)(8,15,18,21)(9,13,10,19)(12,16,20,17)(22,23)(24,30)(25,33)(26,28)(27,31)(29,35)(32,37)(34,36);(6,8,9,16,11,18,10,17)(7,12,19,21,14,20,13,15);(22,24,26,32)(23,27,28,36)(25,34,35,31)(29,37,33,30);(6,9,11,10)(7,13,14,19)(8,16,18,17)(12,15,20,21)(23,28)(27,36)(29,33)(30,37)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12253);