Gene expression data usually comprise small number of samples with tens of thousands of genes. There are a large number of genes unrelated to diseases in this kind of data. The primary task is to detect those key essential genes when analyzing this kind of data. The common feature selection algorithms depend on labels of data, but it is very difficult to get labels for data. To overcome the challenges, especially for gene expression data, the unsupervised feature selection idea is proposed, named as FSSC (feature selection by spectral clustering). FSSC groups all of features into clusters by a spectral clustering algorithm, so that similar features are in same clusters. The feature discernibility and independence are defined, and the feature importance is defined as the product of its discernibility and independence. The representative feature is selected from each cluster to construct the feature subset. According to the spectral clustering algorithms used in FSSC, three kinds of unsupervised feature selection algorithms named as FSSC-SD (FSSC based on standard deviation), FSSC-MD (FSSC based on mean distance) and FSSC-ST (FSSC based on self-tuning) are developed. The SVM (support vector machines) and KNN (K-nearest neighbors) classifiers are adopted to test the performance of the selected feature subsets in experiments. Experimental results on 10 gene expression datasets show that FSSC-SD, FSSC-MD, and FSSC-ST algorithms can select powerful features to classify samples.