  We present high-pass filtered NACO and SINFONI images of the newly discovered
stars S4711-S4715 between 2004 and 2016. Our deep H+K-band (SINFONI) and K-band
(NACO) data show the S-cluster star S4711 on a highly eccentric trajectory
around Sgr~A* with an orbital period of 7.6 years and a periapse distance of
144 AU to the super massive black hole (SMBH). S4711 is hereby the star with
the shortest orbital period and the smallest mean distance to the SMBH during
its orbit to date. The used high-pass filtered images are based on co-added
data sets to improve the signal to noise. The spectroscopic SINFONI data let us
determine detailed stellar properties of S4711 like the mass and the rotational
velocity. The faint S-cluster star candidates, S4712-S4715, can be observed in
a projected distance to Sgr~A* of at least temporarily $\leq$ 120 mas. From
these stars, S4714 is the most prominent one with an orbital period of 12 years
and an eccentricity of 0.985. The stars S4712-S4715 show similar properties
with comparable magnitudes and stellar masses to S4711. The MCMC simulations
determine confidently precise uncertainties for the orbital elements of S62 and
S4711-S4715. The presence of S4711 in addition to S55, S62, and the also newly
found star S4714 implies a population of faint stars that can be found at
distances to Sgr~A* that are comparable to the size of our solar system. These
short orbital time period stars in the dense cluster around the SMBH in the
center of our Galaxy are perfect candidates to observe gravitational effects
such as the periapse shift.
