Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#17760 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C42 | C24×C5 |
| prime radical | C2×C5×#41 | D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C4×C5×#41 ◁ G | |
896 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,9,13,10,11,16,18)(8,12,15,19,14,17,20,21)(23,27)(24,30)(25,32)(28,35)(29,33)(34,36);(7,11)(9,16)(12,17)(15,20)(22,23,26,29)(24,28,30,36)(25,33,32,27)(31,35,37,34);(6,8)(7,12)(9,15)(10,14)(11,17)(13,19)(16,20)(18,21)(22,24)(23,28)(25,31)(26,30)(27,34)(29,36)(32,37)(33,35);(8,14)(12,17)(15,20)(19,21)(24,30)(28,36)(31,37)(34,35)245 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10748);