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

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