Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17386 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#17390 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#121 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C4×C5×Q8 ◁ G | |
940 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,9,12,10,13,16,19)(8,11,15,18,14,17,20,21)(22,23,27,32,28,31,26,30)(24,33,34,37,25,29,35,36);(6,8)(7,11)(9,15)(10,14)(12,18)(13,17)(16,20)(19,21)(22,24)(23,29)(25,28)(26,34)(27,35)(30,36)(31,33)(32,37);(8,14)(11,17)(15,20)(18,21)(22,25,28,24)(23,29,31,33)(26,34,27,35)(30,37,32,36);(22,26,28,27)(23,30,31,32)(24,34,25,35)(29,36,33,37)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10378);