I once used the following MAGMA-code (just for *finite groups*, I don't know if it works for general reductive groups).

    intrinsic Presentation(R::RngInvar) -> SeqEnum
       {A presentation of the invariant ring R.}

	   fund := FundamentalInvariants(R);
	   prim := PrimaryInvariants(R);
	   sec := IrreducibleSecondaryInvariants(R);
	   invar := prim cat sec;
	   P := PolynomialRing(BaseRing(R), #fund);
	   A := Algebra(R);
	
	   invarpres := [];
	   for f in invar do
		   b,g := HomogeneousModuleTest(fund,[R!1],f);
		   Append(~invarpres, g[1]);
	   end for;
	
	   rel := RelationIdeal(R);
	
	   phi:=hom<A->P|invarpres>;
	
	   return ideal<P|[phi(r) : r in Basis(rel)]>;

    end intrinsic;

Put this into a file, say "pres.m", and then you can do:

    > Attach("pres.m");
    > G:=MatrixGroup<2,Rationals() | [-1,0,0,-1] >;
    > G;
    MatrixGroup(2, Rational Field)
    Generators:
    [-1  0]
    [ 0 -1]
    > R:=InvariantRing(G);
    > Presentation(R);
    Ideal of Polynomial ring of rank 3 over Rational Field
    Order: Lexicographical
    Variables: $.1, $.2, $.3
    Homogeneous
    Basis:
    [
    $.1*$.3 - $.2^2
    ]

This is an example of the $A_2$-singularity