Context. As part of the data processing for Gaia Data Release 1 (Gaia DR1) a special astrometric solution was computed, the so-called auxiliary quasar solution. This gives positions for selected extragalactic objects, including radio sources in the second realisation of the International Celestial Reference Frame (ICRF2) that have optical counterparts bright enough to be observed with Gaia. A subset of these positions was used to align the positional reference frame of Gaia DR1 with the ICRF2. Although the auxiliary quasar solution was important for internal validation and calibration purposes, the resulting positions are in general not published in Gaia DR1. Aims. We describe the properties of the Gaia auxiliary quasar solution for a subset of sources matched to ICRF2, and compare their optical and radio positions at the sub-mas level. Methods. Descriptive statistics are used to characterise the optical data for the ICRF sources and the optical-radio differences. The most discrepant cases are examined using online resources to find possible alternative explanations than a physical optical-radio offset of the quasars. Results. In the auxiliary quasar solution 2191 sources have good optical positions matched to ICRF2 sources with high probability. Their formal standard errors are better than 0.76 milliarcsec (mas) for 50% of the sources and better than 3.35 mas for 90%. Optical magnitudes are obtained in Gaia's unfiltered photometric G band. The Gaia results for these sources are given as a separate table in Gaia DR1. The comparison with the radio positions of the defining sources shows no systematic differences larger than a few tenths of a mas. The fraction of questionable solutions, not readily accounted for by the statistics, is less than 6%. Normalised differences have extended tails requiring case-by-case investigations for around 100 sources, but we have not seen any difference indisputably linked to an optical-radio offset in the sources. Conclusions. With less than a quarter of the data expected from the nominal mission it has been possible to obtain positions at the sub-mas level for most of the ICRF sources having an optical counterpart brighter than 20.5 mag.