  It is well known that the generalized Ces\`aro matrices of order one are
hyponormal operators on $\ell^2$, and it has recently been shown that the
Ces\`aro matrix of order two is also hyponormal. Here the relatively new
concept of supraposinormality is used to show that the generalized Ces\`aro
matrices of order two are both posinormal and coposinormal, and that "most" of
them are also hyponormal. A conjecture is propounded that would extend the
hyponormality result.
