Finite Group DatabasebetaFinite Group Databasebeta| order | 15625 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C25 · Schur multiplier (H2) = C52 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C53 | H(3,𝔽5) |
| commutator | C53 | C5×C25 |
| Frattini, prime radical, cosocle | C54 | C52 |
| LCS | G ▷ C53 ▷ C52 ▷ C1 | |
| UCS | C1 ◁ C53 ◁ C54 ◁ G | |
| LECS | G ▷ C54 ▷ C52 ▷ C1 | |
297 conjugacy classes of subgroups.
Polycyclic presentation:
5^6|a^5=f^4,b^5=d,b^a=bc,c^a=ce,c^b=cf745 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15625, 149);