The closed-form displacement analysis of one kind of planar fifth-class groups with 8 links and 12 lower pairs is solved by the method of combining vector with complex number. First, the vector equations are established according to different loops. Then the equations are changed into the form of complex number. The problem is reduced to a single unknown equation after algebraic elimination. The input-output equation, which is proved to be a 42th degree polynomial equation, is obtained. It shows that this construction of fifth-class group has 42 structures. Last, a numerical example is studied. Four real roots are listed. The symbolic computation is carried out by computer algebra system Mathematica.