We report $^{63}$Cu- and $^{205}$Tl-NMR studies on six-layered ($n$=6) high-$T_c$ superconducting (SC) cuprate TlBa$_2$Ca$_5$Cu$_6$O$_{14+delta}$ (Tl1256) with $T_csim$100 K, which reveal that antiferromagnetic (AFM) order takes place below $T_{rm N}sim$170 K. In this compound, four underdoped inner CuO$_2$ planes ($n$(IP)=4) sandwiched by two outer planes (OPs) are responsible for the onset of AFM order, whereas the nearly optimally-doped OPs responsible for the onset of bulk SC. It is pointed out that an increase in the out-of-plane magnetic interaction within an intra-unit-cell causes $T_{rm N}sim$ 45 K for Tl1245 with $n$(IP)=3 to increase to $sim$170 K for Tl1256 with $n$(IP)=4. It is remarkable that the marked increase in $T_{rm N}$ and the AFM moments for the IPs does not bring about any reduction in $T_c$, since $T_csim 100$ K is maintained for both compounds with nearly optimally doped OP. We highlight the fact that the SC order for $nge5$ is mostly dominated by the long-range in-plane SC correlation even in the multilayered structure, which is insensitive to the magnitude of $T_{rm N}$ and the AFM moments at the IPs or the AFM interaction among the IPs. These results demonstrate a novel interplay between the SC and AFM orders when the charge imbalance between the IPs and OP is significantly large.
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