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We precisely determined an $\bN$-graded structure of Zhu's poisson algebra$V/C_2(V)$ of vertex operator algebras $V$ of moonshine type. Namely, if $V$ isa vertex operator algebra of moonshine type with a central charge $24$, then$C_2(V)=\sum_{n=5}^{\infty}V_n+L(-1)V$.

Borcherds's Lie algebra and $C_2$-cofiniteness of vertex operator algebras of moonshine type