Can the executor which is the bank sell the house without notifying the children and in the will it states that it should stay a family house...

If the will states that the executor should try to keep the house in the family, then they have to offer it to the family members to purchase it. It could be that the estate doesn't have enough cash to cover the obligations of the state and needs to sell the house for the cash, but if that's the case, the family should be given the opportunity to buy it.

An executor, however, has broad powers to do what they think best to finalize the estate, but they have to go by the instructions in the will if possible.