Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C210 |
| direct product | C5×#60387 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#120 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| socle | C2×C5 | C22×#120 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
4764 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2,3,4,5)(6,7)(8,17)(9,14)(10,16)(11,23)(12,15)(13,19)(18,33)(20,28)(21,30)(22,29)(24,27)(25,31)(26,35)(32,36)(34,37);(6,8)(7,13)(9,18)(10,19)(11,24)(12,20)(14,26)(15,27)(16,28)(17,23)(21,34)(22,36)(25,35)(29,37)(30,33)(31,32);(6,9)(7,14)(8,18)(10,21)(11,25)(12,22)(13,26)(15,29)(16,30)(17,32)(19,34)(20,36)(23,31)(24,35)(27,37)(28,33);(7,15)(13,27)(14,29)(16,23)(17,28)(26,37)(30,31)(32,33);(6,10,12,11)(7,16,15,23)(8,19,20,24)(9,21,22,25)(13,28,27,17)(14,30,29,31)(18,34,36,35)(26,33,37,32)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53375);