6. (10 points) A class plants trees on Tree Planting Day, divided into 3 groups. The first group plants 5 trees per person, the second group plants 4 trees per person, and the third group plants 3 trees per person. It is known that the number of people in the second group is one-third of the sum of the number of people in the first and third groups, and the number of trees planted by the second group is 72 less than the sum of the number of trees planted by the first and third groups. Then, the class has at least $\qquad$ people.
【Analysis】The first group has $x$ people, the second group has $y$ people, and the third group has $z$ people. Because: The number of people in the second group is $\frac{1}{3}$ of the total number of people in the first and third groups, so:
$y=\frac{1}{3}(x+z), 3 y=x+z, 4 y=x+y+z$ Therefore, the total number of people is a multiple of 4.
Because: The number of trees planted by the second group is 72 less than the total number of trees planted by the first and third groups, so:
$$
\begin{array}{l}
4 y=5 x+3 z-72=2 x+3 y-72 \\
4 y=2 x+3(x+z)-72 \\
4 y=2 x+9 y-72
\end{array}
$$

Substituting $y=1,2,3,4,5,6,7,8$ into the above equations, only 8 satisfies the smallest solution. Then, we can find the total number of people.

【Solution】Let the first group have $x$ people, the second group have $y$ people, and the third group have $z$ people, then:
$$
\begin{array}{l}
y=\frac{1}{3}(x+z) \\
3 y=x+z(2) \\
4 y=x+y+z(3)
\end{array}
$$

From (3), we know that the total number of people is a multiple of 4.
So:
$$
\begin{array}{l}
4 y=5 x+3 z-72=2 x+3 y-72 \\
4 y=2 x+3(x+z)-72 \\
4 y=2 x+9 y-72
\end{array}
$$

Substituting $y=1,2,3,4,5,6,7,8$ into (4), only 8 satisfies the smallest solution.
Therefore, the total number of people is: $4 y=4 \times 8=32$ (people)
Answer: The class has at least 32 people.
The answer is: 32.