Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33539 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33539 |
| 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 | |
1130 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)(8,19)(9,23)(10,18)(11,17)(12,15)(13,20)(14,21)(16,26)(22,34)(24,35)(25,29)(27,33)(28,32)(30,36)(31,37);(6,8)(7,13)(9,24)(10,21)(11,28)(12,22)(14,29)(15,30)(16,32)(17,31)(18,35)(19,23)(20,26)(25,36)(27,37)(33,34);(6,8)(7,14)(9,20)(10,28)(11,21)(12,22)(13,29)(15,32)(16,30)(17,34)(18,19)(23,35)(24,26)(25,37)(27,36)(31,33);(7,15)(13,30)(14,32)(16,29)(17,18)(19,34)(23,33)(31,35)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26527);