Finite Group DatabasebetaFinite Group Databasebeta| order | 3125 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C52 · Schur multiplier (H2) = C5 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C52 | H(3,𝔽5) |
| commutator, Frattini, cosocle | C53 | C52 |
| prime radical | C5×H(3,𝔽5) | C5 |
| LCS | G ▷ C53 ▷ C52 ▷ C1 | |
| UCS | C1 ◁ C52 ◁ C53 ◁ G | |
| LECS | G ▷ C53 ▷ C52 ▷ C1 | |
54 conjugacy classes of subgroups.
| transitive degree | μt(G) = 625 |
Polycyclic presentation:
5^5|b^5=d,b^a=bc,c^a=cd,c^b=ce149 conjugacy classes — view the full character table.
| maximal subgroup of | #30626985, #30626990, C5×#15691062, … |
| maximal normal subgroup of | #30626985, #30626990, C5×#15691062, … |
| maximal quotient of | #30627001, C5×#15691062, #30627005, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3125, 5);