There is no such thing as "the Tarski monster". There exists an uncountable set of Tarski monsters, as was first proved by Olshanskii. His Tarski monsters have infinitely generated Schur multipliers as shown in  Section 5 of ["Lacunary hyperbolic groups"][1]. See also the earlier paper by Ashmanov and Olshanskii MR0829100 where a similar statement was proved for the free Burnside groups (the proof is essentially the same). 


  [1]: https://arxiv.org/abs/math/0701365