TPG is the acronym for Time Projection Chamber with GEM amplification, high-granularity hexaboard read-out and FADC electronics. We have constructed a TPG read-out module, called TPG-head, with three GEM foils and a multilayer board, called hexaboard, covered with 710 000 hexagonal pads of 300 mum size. The total active area of this module is a disk of 30 cm diameter. The 710 000 pads are read by three sets of 576 strips dephased by 120 degrees. Each strip is read by a FADC channel with 100 ns sampling time. The module is mounted in a test-bed formed by a cylindrical field cage of 80 cm diameter and 150 cm length inside a solenoidal magnet that operates with a magnetic field up to B=0.7 Tesla. Tests with X-ray sources show an intrinsic spatial resolution of the order of 40 mum. The threshold for transverse momentum measurement of low energy tracks is below 0.1 MeV/c with a magnetic field of 0.07 Tesla. At 0.7 Tesla the intrinsic space point resolution of the chamber is such that the error on the measurement of the transverse momentum is DeltaPtsime0.1 MeV/c in absence of distortions from the drift region