Finite Group DatabasebetaFinite Group Databasebeta| order | 15625 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C52×C25 · Schur multiplier (H2) = C53 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C25 | C5×H(3,𝔽5) |
| commutator | C52 | C52×C25 |
| Frattini, cosocle | C5×C25 | C53 |
| socle | C5 | C25×H(3,𝔽5) |
| prime radical | H(3,𝔽5)⋊C5 | C25 |
| LCS | G ▷ C52 ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C5×C125 ◁ G | |
| LECS | G ▷ C5×C25 ▷ C5 ▷ C1 | |
254 conjugacy classes of subgroups.
Polycyclic presentation:
5^6|a^5=e,e^5=f^4,b^a=bf,c^b=cd,d^c=df745 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15625, 164);