That group is called the "genotype", and yes, you could have
2 or higher if there's 2 or more non-reference alleles, and you can have a pipe
| instead of
/ if the genotype is phased.
GT genotype, encoded as alleles values separated by either of ”/” or “|”, e.g. The allele values are 0 for the reference allele (what is in the reference sequence), 1 for the first allele listed in ALT, 2 for the second allele list in ALT and so on.
For diploid calls examples could be 0/1 or 1|0 etc. For haploid calls, e.g. on Y, male X, mitochondrion, only one allele value should be given. All samples must have GT call information; if a call cannot be made for a sample at a given locus, ”.” must be specified for each missing allele in the GT field (for example ./. for a diploid).
The meanings of the separators are: / : genotype unphased | : genotype phased
See the VCF spec: http://www.internationalgenome.org/wiki/Analysis/vcf4.0/