Context. The second release of Gaia data (Gaia DR2) contains the astrometric parameters for more than half a million quasars which define akinematically non-rotating reference frame in the optical domain. A subset of them have accurate VLBI positions which allow the axes of thereference frame to be aligned with the ICRF radio frame. Aims. We aim to describe the astrometric and photometric properties of the quasars selected to represent Gaia-CRF2, the celestial reference frame of Gaia DR2, and to compare the optical and radio positions for sources with accurate VLBI positions. Methods. Descriptive statistics are used to characterise the overall properties of the quasar sample. Residual rotation and orientation errors and large-scale systematics are quantified by means of expansions in vector spherical harmonics. Positional differences are calculated relative to a prototype version of the forthcoming ICRF3. Results. Gaia-CRF2 is materialised by the positions of a sample of 556 869 sources in Gaia DR2, obtained from a positional cross-match with the ICRF3-prototype and AllWISE AGN catalogues. The sample constitutes a clean, dense, and homogeneous set of extragalactic point sources in the magnitude range G ' 16 to 21 mag with accurately known optical positions. The median positional uncertainty is 0.12 mas for G < 18 mag and 0.5 mas at G = 20 mag. Large-scale systematics are estimated to be in the range 20 to 30 μas. The accuracy claims are supported by the parallaxes and proper motions of the quasars in Gaia DR2. The optical positions for a subset of 2820 sources in common with the ICRF3-prototype show very good overall agreement with the radio positions, but several tens of sources have significantly discrepant positions. Conclusions. Based on less than 40% of the data expected from the nominal Gaia mission, Gaia-CRF2 is the first realisation of a non-rotating global optical reference frame meeting the ICRS prescriptions, i.e. built only on extragalactic sources. In accuracy it matches the current radio frame realised in the ICRF but with a much higher density of sources in all parts of the sky except along the Galactic equator.