The existing shareholders of Goztepe will retain a 30% stake in the club, who were relegated from the top flight last season.
Sport Republic are aiming to build a portfolio of high-influence stakes in soccer clubs, having bought Southampton in January by picking up an 80% stake in the club.
"Being the first foreign investors in Turkish football comes with extra responsibilities and attention," said Rasmus Ankersen, CEO of Sport Republic and the newly appointed Chairman of Goztepe.
"We know it will not be without challenges, but we are committed to our long-term vision of fulfilling the potential of Goztepe."