Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10374 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#10377 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #393 |
| commutator, Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,9,13)(8,16,18,31)(10,24,22,14)(11,15,23,30)(12,27,28,35)(17,25,29,19)(20,33,21,34)(26,37,36,32);(6,7)(8,17,20,12)(9,13)(10,14)(11,15)(16,19,33,35)(18,29,21,28)(22,24)(23,30)(25,34,27,31)(26,32)(36,37);(6,8,9,21)(7,12,13,29)(10,25,22,35)(11,20,23,18)(14,31,24,33)(15,17,30,28)(16,32,34,37)(19,26,27,36)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3365);