The first author thanks the Spanish Ministry MICINN through grant MTM2017-82740-P for their support. The second and third authors were supported by the Royal Society International Exchanges grant IES\R3\170086.

The purpose of this paper is to study the properties of the irrational-slope Thompson's group Fτ introduced by Cleary in [11]. We construct presentations, both finite and infinite, and we describe its combinatorial structure using binary trees. We show that its commutator group is simple. Finally, inspired by the case of Thompson's group F, we define a unique normal form for the elements of the group and study the metric properties for the elements based on this normal form. As a corollary, we see that several embeddings of F in Fτ are undistorted.