My guess is that they come under supervised learning, as we have labelled dataset of images, but I am not sure as there maybe other aspects in GANs which might come into play in the determination of the class of algorithms GAN falls under.
The terms Supervised Learning and Unsupervised Learning predate the invention of the application of artificial networks to a generative and discriminative network pair, which was the first popular generative topology. The existence of labeling is the key distinction between the two. Even partial labeling indicates supervision, as odd as that jargon is, since the supervisor does no learning and the labels are constants.
Based on the original description of the discriminative network in a GAN, that it consumes examples without labels, GANs are a type of unsupervised learning. That fact does not eliminate the use of labels as part of an extension of the original design that generates based on some labeled element in the examples or the use of other labels to indicate the fitness of each image for a class into which generated images are expected to fall.
GANs are unsupervised machine learning algorithms. According to Wikipedia, unsupervised algorithms are :
Unsupervised learning is a branch of machine learning that learns from test data that has not been labelled, classified or categorized
In GAN networks, only training data is provided which is not labelled. The network generates candidates ( generative ) which are evaluated by the discriminator. The network slowly learns from the data given from a latent space.
Suppose, you want to create a GAN network which can make a Monet painting. You just need to feed it some Monet paintings. Here, you are not interested in classifying the painting, but copying/mimicking it.
Hence, there is no need for labels here which makes GAN an unsupervised machine learning algorithm.