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A fourth leading octal digit carries setuid (4), setgid (2) and the sticky bit (1). setuid makes a binary run as its owner, which is how passwd can edit the shadow file. setgid on a directory makes new files inherit the directory's group, which is what makes shared project directories work. The sticky bit on /tmp stops users deleting each other's files.
Special bits
mode = special digit (4 setuid + 2 setgid + 1 sticky) followed by the three standard digits
Any flaw in one becomes a root escalation. Modern practice replaces them with fine-grained Linux capabilities, so a program gets only the specific privilege it needs.
New files and subdirectories inherit the directory's group rather than the creating user's primary group, which is essential for collaborative directories.