In this article we study the well-known strong lensing system SDSS J1004+4112 . Not only does it host a large-separation lensed quasar with measured time-delay information , but several other lensed galaxies have been identified as well . A previously developed strong lens inversion procedure that is designed to handle a wide variety of constraints , is applied to this lensing system and compared to results reported in other works . Without the inclusion of a tentative central image of one of the galaxies as a constraint , we find that the model recovered by the other constraints indeed predicts an image at that location . An inversion which includes the central image provides tighter constraints on the shape of the central part of the mass map . The resulting model also predicts a central image of a second galaxy where indeed an object is visible in the available ACS images . We find masses of 2.5 \times 10 ^ { 13 } M _ { \odot } and 6.1 \times 10 ^ { 13 } M _ { \odot } within a radius of 60 kpc and 110 kpc respectively , confirming the results from other authors . The resulting mass map is compatible with an elliptical generalization of a projected NFW profile , with r _ { s } = 58 _ { -13 } ^ { +21 } arcsec and c _ { vir } = 3.91 \pm 0.74 . The orientation of the elliptical NFW profile follows closely the orientation of the central cluster galaxy and the overall distribution of cluster members .