Try this, it’s an online tax withholding calculator from the Australian Taxation Office (ATO):
http://www.ato.gov.au/scripts/taxcalc/calc_standard_hire.asp
Make sure you select, Yes to has a Tax File Number (TFN), yes or no to annual leave loading doesn't matter only $2 difference and importantly Yes to HECS debt.
With above in mind, plugged in $840 per week (with annual leave loading = Y) got $240 tax, meaning $600 net.
The problem with your mate may be this. If you haven't completed an employee declaration (or said no to claiming the tax free threshold, you do this when you have a second job) the employer is obligated to take out more tax.
If you use the same figure above BUT say no to claiming the tax free threshold you come out with tax of $317 meaning $523 which around what you said, which may be rounded up by the employer (which a bit bodgy if you ask me)
So I'd get your mate to check that they ARE claiming the tax free threshold on the employee declaration, to minimise tax being withheld each week.
But remember, it all washes up when you lodge you income tax return, because your tax return reconciles what you did pay in tax and what you should pay. So if your mate paid too much in tax, your mate will get a refund (only problem is paying to much tax per week doesn't help the cashflow much).
A tax smart way to deal with HECS is this, get a loan from a bank to payout your HECS, in the process you will get a 15% discount of the remaining HECS balance, get a tax deduction for the interest paid on the loan used to payout your HECS, plus you avoid additional withholding from your pay packet (in your mates case 4.5% extra tax), you avoid indexation on your HECS debt (which isn't tax deductable) and you are realistically reducing your debt instead of your compulsory HECS repayment on tax return being swallowed by the indexation (meaning in some cases you debt is still increasing even with a repayment).
Only downside is you turn a debt with is contingent on your income level, forcing repayment and is generally not declared to banks thus affecting your borrowing capacity vs real debt which has regular repayments and will affect your borrowing capacity, BUT this argument against falls down to the most extent when you have a high income with a high repayment being required, because you will effectively have a regular and fairly high repayment (maybe be in some case equal or more than a possible real loan repayment).
Another trick is in your tax return, you can reduce your actual HECS repayment (which in made on lodgement of your tax return) by reducing your income in your return using expenses/losses such as self education expenses, in some instances you can reduce your income below the bottom threshold, meaning avoiding HECS altogether (I did this a few years in a row). Only exception is rental losses, you can't use a rental loss to reduce HECS BUT you can use negative geared shares etc.
Back on subject:
I'd complete the calculator in the link above, print it off and get your mate to take to the payroll officer to discuss.
Ps. I'm an accountant by day.
Cheers, good luck!
