There are two ways to design a gear box -

Both start with your Top Speed: Which you have already defined for you.

Your next choice is your first gear, which is generally hill climb force, which is kinda redundant for this. So just ignore that bit.

The variance in the method is your inbetween ratios: You can either go for a constant ratio difference (which is stupid and shit) or you can go for a constant speed difference (which is far far better)

Now, idunno whether you have some strategies on distributing the other gears in class, but here's mine: You know you can use your engine from 8000rpm. When accelerating through the gears, you would shift up at 12000rpm and ideally get 8000rpm in the next gear. In other words, top speed in 5th would be 2/3 of the top speed in 6th. Following that strategy you would get these top speeds:

98.3, 65.5, 43.7, 29.1, 19.4, and 12.9 in first.

with respect, you're talking out of your arse. You've not taken into account anything about the power or torque of the engine. You've done the constant ratio difference thing and got things arse about face. Why would you be doing 147 mph to 220 mph when you've got the least torque at the wheels from the engine, when it's easily possible to be doing part of that with a lower gear ratio.

The 'useful power' thing is a bluff, designed to throw you (it's the second stage, not the first). No car would design a gearbox with a constant ratio difference, it would be undriveable. Get a copy of autocar, and look at the gearbox ratios of the car being road tested. Every car will show the gear ratios getting closer and closer together as you move up through the gearbox.

Here's your strategy: Decide your top gear overall ratio (this is pretty much done for you)

Decide your top speed in 1st.

Equally space the top speed in each of the gears between 1st and 6th.

Work out the ratios of the inbetween gears.

check that at top speed in 1st, when you change, the engine speed is over 8000 in second, and the same for moving from top speed in second to third, etc all the way up to 6th.

if this isn't the case, tune your top speed in 1st. and redo the process.

Now that you have your gear ratios, think of how the gearbox is designed: Remember, the bigger the ratio, the physically bigger the gears have to be (And these gears are going to have to heavy duty to begin with to transmit 500kW of power), any way you can think of REDUCING (say for instance, the primary reduction gear) the gearbox ratios will help save weight in the gearbox as well as inertia in the drivetrain.