Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33404 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33404 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C22×#106 |
| commutator | C8 | C23×C4×C5 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C2×C5×#41 | D4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C5×#41 ◁ C2×C5×#131 ◁ G | |
636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(7,12,14,23)(9,19)(10,21)(13,22,25,33)(15,28,27,24)(17,31)(18,30)(20,32)(26,36,35,34)(29,37);(6,7,9,12,10,15,19,24,11,14,20,23,21,27,32,28)(8,13,17,22,18,26,31,34,16,25,29,33,30,35,37,36);(6,8)(7,13)(9,17)(10,18)(11,16)(12,22)(14,25)(15,26)(19,31)(20,29)(21,30)(23,33)(24,34)(27,35)(28,36)(32,37);(8,16)(13,25)(17,29)(18,30)(22,33)(26,35)(31,37)(34,36)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26392);