We experimentally study the properties of a Sodium spin-1 Bose-Einstein condensate with antiferromagnetic interactions. We investigate the equilibrium phase diagram of the system and find good quantitative agreement with mean-field theory. We further study the behavior of the system at small magnetic fields and small magnetization where mean-field theory  breaks down and anomalously large fluctuations are observed. We show that they can be explained by collective spin fluctuations, that would vanish in the thermodynamic limit but are important due to the small size (atom number~few thousands) of the samples we study. This illustrates on a particular example how fluctuations in small systems are effective to counter symmetry breaking. We use the two first moments (mean and variance) of the population in mz=0 to reveal the evolution of the system from a fragmented condensate to a regular mean-field state as the quadratic Zeeman effect increases. We are able to infer the temperature associated to the spin degree of freedom from this two moments, and independently from the distribution of the mz=0 population. We find good agreement between the two methods.