We combined spatially resolved Chandra data with the ROSAT data to constrain the dark-matter distribution in the galaxy group NGC 5044. Within 250(50)(h-1) kpc the total Mass is found to be similar to 1.6 x 10(13) M circle dot 12% of which constitutes baryonic mass. Within tile inner central regions, the total mass profile exhibits a double structure. typical for groups containing a cD galaxy. Following previous Studies of mostly rich galaxy clusters, we studied in detail the nature Of Such a double structure, whereby we inferred likely interface between the cD galaxy and the surrounding galaxies. For this interesting group, we determined for the first time the galaxy-group interface, which is around 7.5 kpc from the peak of the X-ray emission. The total mass internal to this interface radius is found to be similar to 7.1 x 10(10) M circle dot. Beyond this radius, the total mass profile becomes DM-dominated and the corresponding DM profile is reasonably fitted with the NFW model, yielding results consistent with the observed scatter expected for CDM halos. A power-law fit to the DM mass profile gives alpha = 1.88 +/- 0.32, a slope that is within the observed range, but is significantly larger than that of low Surface brightness galaxies and self-interacting DM halos.