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×#33541 |
| 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 | |
1104 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)(9,11)(10,21)(14,29)(15,22)(16,27)(17,28)(19,34)(20,25)(23,24)(26,30)(31,37)(32,35);(6,7)(8,14)(9,15)(10,22)(11,16)(12,17)(13,21)(18,30)(19,35)(20,32)(23,28)(24,27)(25,31)(26,33)(29,34)(36,37);(6,8)(7,14)(9,18)(10,19)(11,25)(12,20)(13,26)(15,30)(16,31)(17,32)(21,33)(22,35)(23,36)(24,34)(27,29)(28,37);(6,9,12,11)(7,15,17,16)(8,18,20,25)(10,21,24,23)(13,27,28,22)(14,30,32,31)(19,33,34,36)(26,29,37,35)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26529);