Relic galaxies are extremely compact early-type objects that have failed to accrete an extended envelope of stars. We observed the relic galaxy NGC 1277 with the integral field George and Cynthia Mitchel spectrograph (GCMS). The observations also include the regular early-type galaxy NGC 1278, which is used for comparison. We obtained resolved kinematics of both targets and performed a Jeans modelling to determine their dark matter content and distribution. We find that, whereas NGC 1278 has a dark matter fraction compatible with the expectations from models, NGC 1277 displays a negligible dark matter fraction within the radius of 6 kpc (five effective radii) from which we recovered stellar kinematics. We propose that the fact that NGC 1277 is dark matter deficient might explain its relic status. This is because for a galaxy lacking an extended dark matter halo, dynamical friction is greatly reduced, resulting in a small efficiency at accreting satellites.