Was the Synagogue of Satan the same as the Synagogue of Libertines? If the Synagogue of Libertines (Acts 6: 9) was Hellenist it would not have been regarded as properly Jewish by conservative Jews. 


  Then there arose certain of the synagogue, which is called the
  synagogue of the Libertines, and Cyrenians, and Alexandrians, and of
  them of Cilicia and of Asia, disputing with Stephen. (Acts 6:9, KJV)


Is this what Revelation 2:9 and Rev. 3: 9 talks about?


  I know thy works, and tribulation, and poverty, (but thou art rich)
  and I know the blasphemy of them which say they are Jews, and are not,
  but are the synagogue of Satan. (Revelation 2:9, KJV)
  
  Behold, I will make them of the synagogue of Satan, which say they are
  Jews, and are not, but do lie; behold, I will make them to come and
  worship before thy feet, and to know that I have loved thee.
  (Revelation 3:9, KJV)
"Synagogue of Satan", according to theologians, may refer to -


Assembly of Jews who rejected Jesus Christ and persecuted the Christians 


  (Pulpit Commentary)
  
  We have here strong evidence of the early date of the Apocalypse.
  Throughout this book "Jew" is an honourable name for the worshippers
  of the Christ; "Gentiles," a name of reproach for those who oppose the
  Christ. [...] These persecutors of the Church of Smyrna are Jews in
  name, but in reality are rather Gentiles - opponents, and not
  worshippers of the Messiah. (source: Bible commentaries on Rev 2:9)

Heretics who professed Christianity


  (Gill's Exposition of the Entire Bible)
  
  they professed Christianity in words, but in works denied it; they
  were men of bad principles and practices, and both blasphemed the ways
  and doctrines of Christ themselves, and caused them to be blasphemed
  by others also; they were false Christians, nominal professors, and
  shunned persecution for the Gospel; who were not what they would be
  thought to be: these were the broachers of heresies in this period of
  time (source: Bible commentaries on Rev 2:9)



"Synagogue of Libertines" was probably the assembly of the Freedmen from Rome, descendants of the Jews enslaved by Pompey after his conquest of Judaea in 63 BC.  (source). They were persecuting the Christians, they were responsible for the death of Stephen, the first martyr. 

Synagogue of Libertines were Hellenistic Jews and were persecuting Christians. If "Synagogue of Satan" means "Jews who reject Christ", then we can deduce that "Synagogue of Libertines" was indeed one of the Synagogues of Satan mentioned in Rev. 2: 9 and Rev. 3: 9.