# Element structure of groups of prime-fifth order

This article gives specific information, namely, element structure, about a family of groups, namely: groups of prime-fifth order.
View element structure of group families | View other specific information about groups of prime-fifth order

## Particular cases

Value of prime $p$ Value of $p^5$ Information on groups of order $p^5$ Information on element structure of groups of order $p^5$ Anomalous?
2 32 groups of order 32 element structure of groups of order 32 Highly
3 243 groups of order 243 element structure of groups of order 243 Highly
5 3125 groups of order 3125 element structure of groups of order 3125 No
7 16807 groups of order 16807 element structure of groups of order 16807 No
11 161051 groups of order 161051 element structure of groups of order 161051 No

The primes $p = 2$ and $p = 3$ behave somewhat differently from the other primes.

## Conjugacy class sizes

FACTS TO CHECK AGAINST FOR CONJUGACY CLASS SIZES AND STRUCTURE:
Divisibility facts: size of conjugacy class divides order of group | size of conjugacy class divides index of center | size of conjugacy class equals index of centralizer
Bounding facts: size of conjugacy class is bounded by order of derived subgroup
Counting facts: number of conjugacy classes equals number of irreducible representations | class equation of a group

### Grouping by conjugacy class sizes

Number of conjugacy classes of size 1 Number of conjugacy classes of size $p$ Number of conjugacy classes of size $p^2$ Number of conjugacy classes of size $p^3$ Total number of conjugacy classes Nilpotency class(es) attained by these Description of groups Number of groups case $p = 2$ Number of groups case $p = 3$ Number of groups case $p \ge 5$
$p^5$ 0 0 0 $p^5$ 1 all abelian groups of order $p^5$ 7 7 7
$p^3$ $p^4 - p^2$ 0 0 $p^4 + p^3 - p^2$ 2 15 15 15
$p$ $p^4 - 1$ 0 0 $p^4 + p - 1$ 2 the extraspecial groups 2 2 2
$p^2$ $p^3 - p$ $p^3 - p^2$ 0 $2p^3 - p$ 2, 3 19 24 $p + 21$
$p^2$ 0 $p^3 - 1$ 0 $p^3 + p^2 - 1$ 3 0 7 $p + 7$
$p$ $p^3 - 1$ 0 $p^2 - p$ $p^3 + p^2 - 1$ 4 3 3 residue class-dependent (fillin)
$p$ $p^2 - 1$ $p^3 - p$ 0 $p^3 + p^2 - 1$ 3 5 6 6
$p$ $p - 1$ $p^2 - 1$ $p^2 - p$ $2p^2 + p - 2$ 4 0 3 6

### Grouping by cumulative conjugacy class sizes (number of elements)

Number of elements in conjugacy classes of size 1 Number of elements in conjugacy classes of size dividing $p$ Number of elements in conjugacy classes of size $p^2$ Number of elements in conjugacy classes of size $p^3$ Total number of conjugacy classes Nilpotency class(es) attained by these Description of groups Number of groups case $p = 2$ Number of groups case $p = 3$ Number of groups case $p \ge 5$
$p^5$ $p^5$ $p^5$ $p^5$ $p^5$ 1 all abelian groups of order $p^5$ 7 7 7
$p^3$ $p^5$ $p^5$ $p^5$ $p^4 + p^3 - p^2$ 2 15 15 15
$p$ $p^5$ $p^5$ $p^5$ $p^4 + p - 1$ 2 the extraspecial groups 2 2 2
$p^2$ $p^4$ $p^5$ $p^5$ $2p^3 - p$ 2, 3 19 24 $p + 21$
$p^2$ $p^2$ $p^5$ $p^5$ $p^3 + p^2 - 1$ 3 0 7 $p + 7$
$p$ $p^4$ $p^4$ $p^5$ $p^3 + p^2 - 1$ 4 3 3 residue class-dependent (fillin)
$p$ $p^3$ $p^5$ $p^5$ $p^3 + p^2 - 1$ 3 5 6 6
$p$ $p^2$ $p^4$ $p^5$ $2p^2 + p - 2$ 4 0 3 6

Note that it is true in this case that the number of elements in conjugacy classes of size dividing any number itself divides the order of the group (in particular, all these numbers are powers of $p$). However, this is not true for all groups and in fact an analogous statement fails for groups of prime-sixth order (see element structure of groups of prime-sixth order). For more, see:

### Correspondence between degrees of irreducible representations and conjugacy class sizes

For groups of order $p^5$, it is true that the list of conjugacy class sizes determines the degrees of irreducible representations. In the case $p = 2$, the converse also holds, i.e., the degrees of irreducible representations determine the conjugacy class sizes. However, for $p \ge 3$, there is one ambiguous case: the case of $p^2$ degree one and $p^3 - 1$ degree two representations corresponds to two possible lists of conjugacy class sizes: ($p$ of size one, $p^3 - 1$ of size $p$, $p^2 - p$ of size $p^3$), and ($p^2$ of size 1, $p^3 - 1$ of size $p^2$). For $p = 2$, there are no groups fitting the latter case.

Number of conjugacy classes of size 1 Number of conjugacy classes of size $p$ Number of conjugacy classes of size $p^2$ Number of conjugacy classes of size $p^3$ Total number of conjugacy classes = number of irreducible representations Number of degree 1 irreps Number of degree $p$ irreps Number of degree $p^2$ irreps
$p^5$ 0 0 0 $p^5$ $p^5$ 0 0
$p^3$ $p^4 - p^2$ 0 0 $p^4 + p^3 - p^2$ $p^4$ $p^3 - p^2$ 0
$p$ $p^4 - 1$ 0 0 $p^4 + p - 1$ $p^4$ 0 $p - 1$
$p^2$ $p^3 - p$ $p^3 - p^2$ 0 $2p^3 - p$ $p^3$ $p^3 - p$ 0
$p^2$ 0 $p^3 - 1$ 0 $p^3 + p^2 - 1$ $p^2$ $p^3 - 1$ 0
$p$ $p^3 - 1$ 0 $p^2 - p$ $p^3 + p^2 - 1$ $p^2$ $p^3 - 1$ 0
$p$ $p^2 - 1$ $p^3 - p$ 0 $p^3 + p^2 - 1$ $p^3$ $p^2 - p$ $p - 1$
$p$ $p - 1$ $p^2 - 1$ $p^2 - p$ $2p^2 + p - 2$ $p^2$ $p^2 - 1$ $p - 1$