Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33856 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33858 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | D4×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C5×#443 ◁ G | |
668 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)(7,13)(8,18)(9,22)(10,11)(14,15)(16,32)(17,31)(19,36)(20,34)(21,27)(23,28)(25,26)(29,30)(33,35);(6,7,12,17)(8,19,21,14)(9,23,26,32)(10,16,11,28)(13,25,31,22)(15,34,36,20)(18,33,27,30)(24,35,37,29);(6,8)(7,14)(9,20)(10,27)(11,18)(12,21)(13,29)(15,32)(16,33)(17,19)(22,24)(23,36)(25,37)(26,34)(28,30)(31,35);(6,8)(7,15)(9,24)(10,18)(11,27)(12,21)(13,30)(14,32)(16,35)(17,36)(19,23)(20,22)(25,34)(26,37)(28,29)(31,33)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26846);