One-bit compressed sensing (1-bit CS) shows that a sparse signal can exactly be recovered from extremely quantized linear measurements which just catch their signs. While in many applications, real-world signals also exhibit additional structures aside from standard sparsity, such as block-sparsity. We proposed a nonconvex ℓ₂ / ℓ_p ( 0 < p < 1 ) -minimization model. Using the best approximation, covering number, and packing number, we obtain a weak condition to reconstruct block-sparse signals with high probability. And the lower bound of the required number of measurements is lower than some existing 1-bit CS methods. At last, we propose a block adaptive binary iteration thresholding algorithm to recover ℓ₂ / ℓ_p effectively block sparse signals. The algorithm can be used without knowing the sparsity of the underlying signal. Several simulations are conducted to reveal the superiority of our methods to existing approaches, which expose the advantage of 1-bit CS in the reconstruction of block-sparse signals.