Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33531 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33531 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×#447 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
1122 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)(9,19)(10,22)(13,27)(14,26)(15,30)(17,32)(18,31)(20,24)(21,35)(23,36)(25,28)(29,37)(33,34);(6,7)(8,13)(9,20)(10,24)(11,14)(12,19)(15,22)(16,28)(17,33)(18,34)(21,37)(23,30)(25,35)(26,36)(27,32)(29,31);(6,8)(7,13)(9,21)(10,18)(11,16)(12,25)(14,28)(15,29)(17,23)(19,35)(20,37)(22,31)(24,34)(26,27)(30,33)(32,36);(8,16)(13,28)(17,21)(18,31)(25,27)(29,34)(32,35)(33,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26519);