Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33657 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33659 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×#481 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C5×#3060 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
546 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)(6,7)(8,17)(9,14)(10,23)(11,21)(12,16)(13,20)(15,26)(18,32)(19,34)(22,24)(25,31)(27,37)(28,35)(29,36)(30,33);(6,7)(8,13)(9,21)(10,24)(11,14)(12,16)(15,27)(17,33)(18,31)(19,30)(20,34)(22,23)(25,35)(26,37)(28,29)(32,36);(6,8,11,28,12,19,9,18)(7,13,14,29,16,30,21,31)(10,25,26,33,27,36,22,20)(15,32,23,34,24,35,37,17);(6,9,12,11)(7,14,16,21)(8,18,19,28)(10,22,27,26)(13,29,30,31)(15,23,24,37)(17,32,34,35)(20,36,33,25)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26647);