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×#33533 |
| 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 | |
1102 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)(13,19)(15,26)(18,34)(20,35)(22,25)(24,29)(27,36)(28,32)(30,33)(31,37);(6,8)(7,13)(9,18)(10,24)(11,28)(12,20)(14,29)(15,32)(16,30)(17,23)(19,26)(21,31)(22,33)(25,34)(27,37)(35,36);(6,7)(8,13)(9,21)(10,25)(11,14)(12,16)(15,27)(17,33)(18,31)(19,35)(20,30)(22,23)(24,34)(26,36)(28,29)(32,37);(6,9,12,11)(7,14,16,21)(8,18,20,28)(10,22,27,26)(13,29,30,31)(15,23,25,36)(17,34,35,32)(19,24,33,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26521);