Identification and delineation of craniofacial characteristics support the clinical and molecular diagnosis of genetic syndromes. Deep learning (DL) frameworks for syndrome identification from 2D facial images are trained on large clinical datasets using standard convolutional neural networks for classification. In contrast, despite the increased availability of 3D scanners in clinical setups, similar frameworks remain absent for 3D facial photographs. The main challenges involve working with smaller datasets and the need for DL operations applicable to 3D geometric data. Therefore, to date, most 3D methods refrain from working across multiple syndromic groups and/or are solely based on traditional machine learning. The first contribution of this work is the use of geometric deep learning with spiral convolutions in a triplet-loss architecture. This geometric encoding (GE) learns a lower dimensional metric space from 3D facial data that is used as input to linear discriminant analysis (LDA) performing multiclass classification. Benchmarking is done against principal component analysis (PCA), a common technique in 3D facial shape analysis, and related work based on 65 distinct 3D facial landmarks as input to LDA. The second contribution of this work involves a part-based implementation to 3D facial shape analysis and multi-class syndrome classification, and this is applied to both GE and PCA. Based on 1,786 3D facial photographs of controls and individuals from 13 different syndrome classes, a five-fold cross-validation was used to investigate both contributions. Results indicate that GE performs better than PCA as input to LDA, and this especially so for more compact (lower dimensional) spaces. In addition, a part-based approach increases performance significantly for both GE and PCA, with a more significant improvement for the latter. I.e., this contribution enhances the power of the dataset. Finally, and interestingly, according to ablation studies within the part-based approach, the upper lip is the most distinguishing facial segment for classifying genetic syndromes in our dataset, which follows clinical expectation. This work stimulates an enhanced use of advanced part-based geometric deep learning methods for 3D facial imaging in clinical genetics.