# Groups of order 243

## Contents

See pages on algebraic structures of order 243| See pages on groups of a particular order

## Statistics at a glance

Since $243 = 3^5$ is a prime power, and prime power order implies nilpotent, all groups of order 243 are nilpotent groups.

Quantity Value Explanation
Total number of groups up to isomorphism 67
Number of abelian groups 7 Equals the number of unordered integer partitions of 5, see classification of finite abelian groups
Number of groups of nilpotency class exactly two 28
Number of groups of nilpotency class exactly three 26
Number of groups of nilpotency class exactly four (i.e., maximal class groups) 6

## Arithmetic functions

### Summary information

Here, the rows are arithmetic functions that take values between $0$ and $5$, and the columns give the possible values of these functions. The entry in each cell is the number of isomorphism classes of groups for which the row arithmetic function takes the column value. Note that all the row value sums must equal $67$, which is the total number of groups of order $243$

Arithmetic function Value 0 Value 1 Value 2 Value 3 Value 4 Value 5
prime-base logarithm of exponent 0 4 49 11 2 1
Frattini length 0 1 46 17 2 1
nilpotency class 0 7 28 26 6 0
derived length 0 7 60 0 0 0
minimum size of generating set 0 1 29 30 6 1
rank as p-group 0 1 15 42 8 1
normal rank as p-group 0 1 15 42 8 1
characteristic rank as p-group 0 1 17 39 7 1