The structure and representation of nilpotent Lie algebra play an important role in the Lie theory.The algebraic structure of a certain class of six-dimensional nilpotent Lie algebras with the parameter ε over the complex field was discussed.It is determined that in the sense of isomorphism,the automorphism group of this class of six-dimensional nilpotent Lie algebra is isomorphic to a six-order matrix multiplication group by using the definition and properties of isomorphism and a large amount of calculation.Then the properties of Centroid algebras of this class of six-dimensional nilpotent Lie algebra were analysed and its matrix representation was given.It is shown that the Centroid algebra is a six-dimensional nilpotent Lie algebra.Finally,the δ-derivation of this class of six-dimensional nilpotent Lie algebras was determined.Especially in the case of δ=1,the structure of derivation algebras was discussed and it is concluded that the derivation algebra is a ten-dimensional Lie algebra and outer derivation algebra is a five-dimensional Lie algebra.