  Samples of galaxy clusters allow us to better understand the physics at play
in galaxy formation and to constrain cosmological models once their mass,
position (for clustering studies) and redshift are known. In this context,
large optical data sets play a crucial role. We investigate the capabilities of
the Javalambre-Physics of the Accelerating Universe Astrophysical Survey
(J-PAS) in detecting and characterizing galaxy groups and clusters. We analyze
the data of the miniJPAS survey, obtained with the JPAS-Pathfinder camera and
covering $1$ deg$^2$ centered on the AEGIS field to the same depths and with
the same 54 narrow band plus 2 broader band near-UV and near-IR filters
anticipated for the full J-PAS survey. We use the Adaptive Matched Identifier
of Clustered Objects (AMICO) to detect and characterize groups and clusters of
galaxies down to $S/N=2.5$ in the redshift range $0.05<z<0.8$. We detect 80, 30
and 11 systems with signal-to-noise ratio larger than 2.5, 3.0 and 3.5,
respectively, down to $\sim 10^{13}\,M_{\odot}/h$. We derive mass-proxy scaling
relations based on Chandra and XMM-Newton X-ray data for the signal amplitude
returned by AMICO, the intrinsic richness and a new proxy that incorporates the
galaxies' stellar masses. The latter proxy is made possible thanks to the J-PAS
filters and shows a smaller scatter with respect to the richness. We fully
characterize the sample and use AMICO to derive a probabilistic membership
association of galaxies to the detected groups that we test against
spectroscopy. We further show how the narrow band filters of J-PAS provide a
gain of up to 100% in signal-to-noise ratio in detection and an uncertainty on
the redshift of clusters of only $\sigma_z=0.0037(1+z)$ placing J-PAS in
between broadband photometric and spectroscopic surveys. The performances of
AMICO and J-PAS with respect to mass sensitivity, mass-proxies quality
