Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#10199 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C23×C5 | #127 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,12,17)(8,18,9,20)(10,16,11,15)(13,21,14,19)(24,31)(25,26)(28,32)(29,35)(33,34)(36,37);(6,8,12,9)(7,13,17,14)(10,21,11,19)(15,20,16,18)(22,23)(24,32)(25,29)(26,35)(27,30)(28,34)(31,36)(33,37);(6,9,12,8)(7,14,17,13)(10,21,11,19)(15,20,16,18)(22,24)(23,28)(25,34)(26,31)(27,33)(29,37)(30,36)(32,35)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3187);