The invention relates to the wheat cultivar designated 01071782. Provided by the invention are the seeds, plants and derivatives of the wheat cultivar 01071782. Also provided by the invention are tissue cultures of the wheat cultivar 01071782 and the plants regenerated therefrom. Still further provided by the invention are methods for producing wheat plants by crossing the wheat cultivar 01071782 with itself or another wheat cultivar and plants produced by such methods.