Explicitly using the block structure of the unknown signal can achieve better reconstruction performance in compressive sensing. Theoretically, an unknown signal with block structure can be accurately recovered from a few number of under-determined linear measurements provided that it is sufficiently block sparse. From the practical point of view, a severe concern is that the block sparse level appears often unknown. In this paper, we introduce a soft measure of block sparsity $k_α(\mathbf{x})=\left(\lVert\mathbf{x}\rVert_{2,α}/\lVert\mathbf{x}\rVert_{2,1}\right)^{\fracα{1-α}}$ with $α\in[0,\infty]$, and propose an estimation procedure by using multivariate centered isotropic symmetric $α$-stable random projections. The limiting distribution of the estimator is established. Simulations are conducted to illustrate our theoretical results.