Finite Group DatabasebetaFinite Group Databasebeta| order | 15625 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C53 · Schur multiplier (H2) = C55 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C5 | C52⋊H(3,𝔽5) |
| commutator, Frattini, cosocle | C53 | C53 |
| prime radical | C52⋊H(3,𝔽5) | C5 |
| LCS | G ▷ C53 ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C5 ◁ C53 ◁ G | |
| LECS | G ▷ C53 ▷ C5 ▷ C1 | |
756 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 756 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
5^6|a^5=f^4,b^5=f^4,c^a=cd,d^a=df,c^b=ce,e^b=ef^4265 conjugacy classes — view the full character table.
| derived subgroup of | #72229808, #72229822 |
| Fitting subgroup of | #72229808, #72229822 |
| maximal subgroup of | #72229808 |
| maximal normal subgroup of | #72229808 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15625, 612);