Abstract: We describe explicitly non-negative extremals for the Sobolev inequality on the seven dimensional quaternionic Hesenberg group and determine the best constant in the $L^2$ Folland-Stein embedding theorem involving quaternionic contact geometry and the Biquard connection. Consequently, we obtain a solution of the quaternionic contact Yamabe problem on the quaternionic seven sphere. We show that the torsion of the Biquard connection is the only obstruction quaternionic contact structure to be locally isomorphic to a 3-Sasakian one. We define a curvature-type tensor invariant called quaternionic contact (qc) conformal curvature in terms of the curvature and torsion of the Biquard connection. The discovered tensor is similar to the Weyl conformal curvature in Riemannian geometry and to the Chern-Moser invariant in CR geometry. We show that a quaternionic contact manifold is locally qc conformal (gauge equivalent) to the standard flat quaternionic contact structure on the quaternionic Heisenberg group, or equivalently, to the standard 3-sasakian structure on the sphere if and only if the qc conformal curvature vanishes.