Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#22694 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | D42 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | D4×C4⋊C4 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ G | |
812 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)(6,7,12,17,13,18,11,30)(8,19,21,14,23,32,22,33)(9,24,20,34,26,31,25,15)(10,16,27,36,29,37,28,35);(6,8)(7,14)(9,10)(11,22)(12,21)(13,23)(15,16)(17,32)(18,33)(19,30)(20,27)(24,36)(25,28)(26,29)(31,35)(34,37);(6,9,11,25,13,26,12,20)(7,15,17,24,18,34,30,31)(8,10,22,28,23,29,21,27)(14,16,32,36,33,37,19,35);(6,10)(7,16)(8,20)(9,22)(11,28)(12,27)(13,29)(14,31)(15,32)(17,36)(18,37)(19,34)(21,26)(23,25)(24,33)(30,35)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15682);