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We present high-pass filtered NACO and SINFONI images of the newly discoveredstars 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 trajectoryaround Sgr~A* with an orbital period of 7.6 years and a periapse distance of144 AU to the super massive black hole (SMBH). S4711 is hereby the star withthe shortest orbital period and the smallest mean distance to the SMBH duringits orbit to date. The used high-pass filtered images are based on co-addeddata sets to improve the signal to noise. The spectroscopic SINFONI data let usdetermine detailed stellar properties of S4711 like the mass and the rotationalvelocity. The faint S-cluster star candidates, S4712-S4715, can be observed ina projected distance to Sgr~A* of at least temporarily $\leq$ 120 mas. Fromthese stars, S4714 is the most prominent one with an orbital period of 12 yearsand an eccentricity of 0.985. The stars S4712-S4715 show similar propertieswith comparable magnitudes and stellar masses to S4711. The MCMC simulationsdetermine confidently precise uncertainties for the orbital elements of S62 andS4711-S4715. The presence of S4711 in addition to S55, S62, and the also newlyfound star S4714 implies a population of faint stars that can be found atdistances to Sgr~A* that are comparable to the size of our solar system. Theseshort orbital time period stars in the dense cluster around the SMBH in thecenter of our Galaxy are perfect candidates to observe gravitational effectssuch as the periapse shift.

S62 and S4711: Indications of a population of faint fast moving stars inside the S2 orbit -- S4711 on a 7.6 year orbit around Sgr~A*