A rogue spaceship escapes. $54$ minutes later the police leave in a spaceship in hot pursuit. If the police spaceship travels $12\%$ faster than the rogue spaceship along the same route, how many minutes will it take for the police to catch up with the rogues?
1. Let's denote the speed of the rogue spaceship as \( v \) units per hour. 
2. The police spaceship travels \( 12\% \) faster than the rogue spaceship, so its speed is \( v + 0.12v = 1.12v \) units per hour.
3. The rogue spaceship has a head start of 54 minutes. Converting this to hours, we get:
   \[
   \frac{54}{60} = 0.9 \text{ hours}
   \]
4. During this time, the rogue spaceship travels:
   \[
   \text{Distance} = v \times 0.9 = 0.9v \text{ units}
   \]
5. The police spaceship needs to cover this distance to catch up. The relative speed at which the police spaceship is closing the gap is:
   \[
   1.12v - v = 0.12v \text{ units per hour}
   \]
6. The time \( t \) it takes for the police spaceship to catch up is given by:
   \[
   t = \frac{0.9v}{0.12v} = \frac{0.9}{0.12} = 7.5 \text{ hours}
   \]
7. Converting this time into minutes:
   \[
   7.5 \text{ hours} \times 60 \text{ minutes per hour} = 450 \text{ minutes}
   \]

The final answer is \(\boxed{450}\) minutes.