  Context. The first Gaia Data Release contains the Tycho-Gaia Astrometric
Solution (TGAS). This is a subset of about 2 million stars for which, besides
the position and photometry, the proper motion and parallax are calculated
using Hipparcos and Tycho-2 positions in 1991.25 as prior information. Aims. We
investigate the scientific potential and limitations of the TGAS component by
means of the astrometric data for open clusters. Methods. Mean cluster parallax
and proper motion values are derived taking into account the error correlations
within the astrometric solutions for individual stars, an estimate of the
internal velocity dispersion in the cluster, and, where relevant, the effects
of the depth of the cluster along the line of sight. Internal consistency of
the TGAS data is assessed. Results. Values given for standard uncertainties are
still inaccurate and may lead to unrealistic unit-weight standard deviations of
least squares solutions for cluster parameters. Reconstructed mean cluster
parallax and proper motion values are generally in very good agreement with
earlier Hipparcos-based determination, although the Gaia mean parallax for the
Pleiades is a significant exception. We have no current explanation for that
discrepancy. Most clusters are observed to extend to nearly 15 pc from the
cluster centre, and it will be up to future Gaia releases to establish whether
those potential cluster-member stars are still dynamically bound to the
clusters. Conclusions. The Gaia DR1 provides the means to examine open clusters
far beyond their more easily visible cores, and can provide membership
assessments based on proper motions and parallaxes. A combined HR diagram shows
the same features as observed before using the Hipparcos data, with clearly
increased luminosities for older A and F dwarfs.
