 Z(G(n)) is finite for n>3.
 [Selick] the pprimary component of Z(G(n)) has exponent p for p>2.
 [JamesCohen] the 2primary component of Z(G(n)) has exponent 4.
 [Toda] Z(G(n)) is known for n<23.
 [CurtisMahowald] the 2primary component of Z(G(n)) is known up to n=64 or
so.
 Here are some examples (from Toda's book, composition methods in
homotopy groups of spheres):
 Z(G(2))=Z(G(3))=Z,
 Z(G(4))=Z/2Z,
 Z(G(5))=Z/2Z,
 Z(G(6))=Z/4Z +Z/3Z,
 Z(G(7))=Z/2Z,
 Z(G(8))=Z/2Z,
 Z(G(9))=Z/3Z,
 Z(G(10))=Z/3Z+Z/5Z,
 Z(G(11))=Z/2Z,
 Z(G(12))=Z/2Z+Z/2Z,
 Z(G(13))=Z/4Z+Z/2Z+Z/3Z,

Z(G(14))=Z/4Z+Z/2Z+Z/2Z+Z/3Z+Z/7Z,
 Z(G(15))=Z/2Z+Z/2Z,
 Z(G(16))=Z/2Z+Z/3Z,
 Z(G(17))=Z/2Z+Z/3Z+Z/5Z,
 Z(G(18))=Z/2Z+Z/3Z+Z/5Z,
 Z(G(19))=Z/2Z+Z/2Z+Z/3Z,

Z(G(20))=Z/4Z+Z/2Z+Z/2Z+Z/3Z,
 Z(G(21))=Z/4Z+Z/2Z+Z/2Z
+Z/3Z,

Z(G(22))=Z/4Z+Z/2Z+Z/3Z+Z/11Z,
Observe that in this table the ptorsion component of Z(G(n)) is either
Z/pZ or 0 for p>2. Also
Z/4Z+Z/2Z+Z/2Z is the maximal 2component
of Z(G(n)) in this table.
The calculations given by Curtis and Mahowald also suggest that Z(G(n)) is
very small in some sense.