Groups of order 3125

This article gives information about, and links to more details on, groups of order 3125
See pages on algebraic structures of order 3125 | See pages on groups of a particular order

Statistics at a glance

To understand these in a broader context, see
groups of prime-fifth order|groups of order 5^n

Since $3125=5^5$ is a prime power, and prime power order implies nilpotent, all groups of order 3125 are nilpotent groups.

Quantity	Value	Explanation
Total number of groups up to isomorphism	77	PORC function for number of groups of order $p^5$ for $p\ge 5$ is: $2p+61+2\mathrm{g}\mathrm{c}\mathrm{d}(p-1,3)+\mathrm{g}\mathrm{c}\mathrm{d}(p-1,4)$. Plugging in $p=5$ gives 77.
Number of abelian groups	7	equals the number of unordered integer partitions of $5$ (this is the $5$ appearing in the exponent part of $5^5$). See classification of finite abelian groups
Number of groups of nilpotency class exactly two	30	the general formula for order $p^5$ with $p\ge 5$ is $p+25$.
Number of groups of nilpotency class exactly three	31	the general formula for order $p^5$ with $p\ge 5$ is $p+26$.
Number of groups of nilpotency class exactly four (i.e., maximal class groups)	9	the general formula for order $p^5$ with $p\ge 5$ is $3+2\mathrm{g}\mathrm{c}\mathrm{d}(p-1,3)+\mathrm{g}\mathrm{c}\mathrm{d}(p-1,4)$.