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This research includes the study of laminar, two dimensional boundary layer flow of MHD Maxwell nanofluid over a vertically moving flat porous plate. The concentration, thermal buoyancy effects and Brownian motion have significant importance in this work. We have considered heat absorbing fluid with viscous dissipation and radiation effects. We have applied appropriate similarity variables to reduce the governing equations into a system of non-linear ordinary differential equations. Simplified form of coupled nonlinear system of ordinary differential equations is solved numerically through numerical package with accuracy. The results are discussed through graphs and tables.