The Generalized Feistel Structure (GFS) is one of the structures used in designs of blockciphers and hash functions. There are several types of GFSs, and we focus on Type 1 and Type 2 GFSs. The security of these structures are well studied and they are adopted in various practical blockciphers and hash functions. The round function used in GFSs consists of two layers. The first layer uses the nonlinear function. Type 1 GFS uses one nonlinear function in this layer, while Type 2 GFS uses a half of the number of sub-blocks. The second layer is a sub-block-wise permutation, and the cyclic shift is generally used in this layer. In this paper, we formalize Type 1.x GFS, which is the natural extension of Type 1 and Type 2 GFSs with respect to the number of nonlinear functions in one round. Next, for Type 1.x GFS using two nonlinear functions in one round, we propose a permutation which has a good diffusion property. We demonstrate that Type 1.x GFS with this permutation has a better diffusion property than other Type 1.x GFS with the sub-block-wise cyclic shift. We also present experimental results of evaluating the diffusion property and the security against the saturation attack, impossible differential attack, differential attack, and linear attack of Type 1.x GFSs with various permutations.