The legal position is that unless the employment says something different, the employee has an obligation to give one week's notice if he/she has worked longer than one month. That is the statutory obligation. If you want to 'keep clean' I would do that - it's not too much of thing to do.
The employer is in breach of another statutory obligation - within two months of your start date, it should have given you written notice of the particualrs of your employement - and it didn't. There's a bunch of minimum things that must be provided - one of which is the details of the notice periods (on each side) that must be provided.
If they start getting shirty and want a month, just remind them of this.