Function 
Value 
Similar groups 
Explanation for function value


underlying prime of pgroup 
2 


order (number of elements, equivalently, cardinality or size of underlying set) 
{{{order}}} 
groups with same order "{{{order}}}" is not a number. 

primebase logarithm of order 
{{{order plog}}} 
groups with same primebase logarithm of order "{{{order plog}}}" is not a number. 

maxlength of a group 
{{{order plog}}} 

maxlength of a group equals primebase logarithm of order for group of prime power order

chief length 
{{{order plog}}} 

chief length equals primebase logarithm of order for group of prime power order

composition length 
{{{order plog}}} 

composition length equals primebase logarithm of order for group of prime power order

exponent of a group 
{{{degree}}} 
groups with same order and exponent of a group "{{{degree}}}" is not a number.
 "{{{order}}}" is not a number.
 groups with same exponent of a group"{{{degree}}}" is not a number. 
cyclic subgroup of order {{{degree}}}.

primebase logarithm of exponent 
{{{degree plog}}} 
groups with same order and primebase logarithm of exponent "{{{degree plog}}}" is not a number.
 "{{{order}}}" is not a number.
 groups with same primebase logarithm of order and primebase logarithm of exponent "{{{degree plog}}}" is not a number.
 "{{{order plog}}}" is not a number.
 groups with same primebase logarithm of exponent"{{{degree plog}}}" is not a number. 

nilpotency class 
2 
groups with same order and nilpotency class "{{{order}}}" is not a number.  groups with same primebase logarithm of order and nilpotency class"{{{order plog}}}" is not a number.  groups with same nilpotency class 

derived length 
2 
groups with same order and derived length "{{{order}}}" is not a number.  groups with same primebase logarithm of order and derived length"{{{order plog}}}" is not a number.  groups with same derived length 
the derived subgroup is contained in the cyclic subgroup and is hence abelian

Frattini length 
{{{degree plog}}} 
groups with same order and Frattini length "{{{degree plog}}}" is not a number.
 "{{{order}}}" is not a number.
 groups with same primebase logarithm of order and Frattini length "{{{degree plog}}}" is not a number.
 "{{{order plog}}}" is not a number.
 groups with same Frattini length"{{{degree plog}}}" is not a number. 

minimum size of generating set 
2 
groups with same order and minimum size of generating set "{{{order}}}" is not a number.  groups with same primebase logarithm of order and minimum size of generating set"{{{order plog}}}" is not a number.  groups with same minimum size of generating set 

subgroup rank of a group 
2 
groups with same order and subgroup rank of a group "{{{order}}}" is not a number.  groups with same primebase logarithm of order and subgroup rank of a group"{{{order plog}}}" is not a number.  groups with same subgroup rank of a group 
All proper subgroups are cyclic, dihedral, or Klein fourgroups.

rank of a pgroup 
2 
groups with same order and rank of a pgroup "{{{order}}}" is not a number.  groups with same primebase logarithm of order and rank of a pgroup"{{{order plog}}}" is not a number.  groups with same rank of a pgroup 
there exist Klein foursubgroups.

normal rank of a pgroup 
2 
groups with same order and normal rank of a pgroup "{{{order}}}" is not a number.  groups with same primebase logarithm of order and normal rank of a pgroup"{{{order plog}}}" is not a number.  groups with same normal rank of a pgroup 
all abelian normal subgroups are cyclic.

characteristic rank of a pgroup 
2 
groups with same order and characteristic rank of a pgroup "{{{order}}}" is not a number.  groups with same primebase logarithm of order and characteristic rank of a pgroup"{{{order plog}}}" is not a number.  groups with same characteristic rank of a pgroup 
There is a unique (hence characteristic) Klein foursubgroup.
