The structure in the cell's nucleus that holds hereditary information is the chromosomes.

Here's a breakdown:

* Chromosomes are long, thread-like structures made of DNA and proteins.

* DNA (deoxyribonucleic acid) contains the genetic instructions that determine an organism's traits.

* Genes are segments of DNA that code for specific traits.

So, the chromosomes within the nucleus house the DNA, which in turn contains the genes that determine an organism's heredity.