Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3026 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#20964 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#120 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#386 |
| prime radical | C5×#3449 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1700 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)(6,7,10,15,11,14,23,28)(8,12,18,26,17,25,32,36)(9,19,21,29,20,13,22,27)(16,24,31,35,30,34,33,37);(6,8,10,18,11,17,23,32)(7,12,15,26,14,25,28,36)(9,16,22,33,20,30,21,31)(13,24,29,37,19,34,27,35);(6,9)(7,13)(8,16)(10,22)(11,20)(12,24)(14,19)(15,29)(17,30)(18,33)(21,23)(25,34)(26,37)(27,28)(31,32)(35,36);(7,14)(12,25)(13,19)(15,28)(24,34)(26,36)(27,29)(35,37)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13952);