Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#9633 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #121 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | #389 |
| prime radical | C23×C5 | #125 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,10,13,11,14,9,18)(8,15,16,12,17,20,19,21)(23,28)(24,31)(25,26)(29,36)(30,34)(32,33);(6,8,10,16,11,17,9,19)(7,12,13,20,14,21,18,15)(22,23)(24,29)(25,34)(26,28)(27,30)(31,35)(32,37)(33,36);(22,24,26,31,27,33,25,32)(23,29,28,35,30,36,34,37)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2621);