I really want a 'Pro- Pure Mortal' group than an anti magic/supernatural group. One that is designed and operates under the assumption that Pure Mortal Bloodlines is worth more than all the supernaturals around. I think a little white supremacy mixed with the finances (and secrecy?) of the Bilderberg Group.
Something similar to the 'Ancient Blood Cult' from the 2nd season of tv series Dark Angel could work nicely for you. In the series this is a pseudo-religious cult who have amassed a vast amount of power and influence throughout their millenia long existence. They are obsessed with breeding 'pure' humanity within their ranks, see the majority of humanity as unclean, or tainted, and the mutants and transgenics in the series as abominations to be purged. In the series, this carefully controlled breeding programme has allowed them to become more than mere humanity, they are capable of taking on genetically engineered supersoldiers in hand to hand combat and wiping the floor with them, and some of their members show psionic aptitude.
You could take this and replace 'transgenic' with 'supernatural beings'. Their stance on Wizards and other magic users could be more difficult to calculate, as magical powers could be considered a great boon, something to be bred to their bloodlines, but might be considered a corrupting, impure taint (perhaps room for an internal division between the wilder, younger members and the conservative leaders).