  Angle-integrated cross-section measurements of the $^{56}$Ni(d,n) and (d,p)
stripping reactions have been performed to determine the single-particle
strengths of low-lying excited states in the mirror nuclei pair
$^{57}$Cu-$^{57}$Ni situated adjacent to the doubly magic nucleus $^{56}$Ni.
The reactions were studied in inverse kinematics utilizing a beam of
radioactive $^{56}$Ni ions in conjunction with the GRETINA $\gamma$-array.
Spectroscopic factors are compared with new shell-model calculations using a
full $pf$ model space with the GPFX1A Hamiltonian for the isospin-conserving
strong interaction plus Coulomb and charge-dependent Hamiltonians. These
results were used to set new constraints on the $^{56}$Ni(p,$\gamma$)$^{57}$Cu
reaction rate for explosive burning conditions in x-ray bursts, where $^{56}$Ni
represents a key waiting point in the astrophysical rp-process.
