Finite Group DatabasebetaFinite Group Databasebeta| order | 3125 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C52 · Schur multiplier (H2) = C5 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C5 | H(3,𝔽5)⋊C5 |
| commutator, Frattini, cosocle | C53 | C52 |
| prime radical | C54 | C5 |
| LCS | G ▷ C53 ▷ C52 ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C5 ◁ C52 ◁ C53 ◁ G | |
| LECS | G ▷ C53 ▷ C52 ▷ C5 ▷ C1 | |
244 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 25 |
Polycyclic presentation:
5^5|a^5=e,b^a=bc,c^a=cd,d^a=de149 conjugacy classes — view the full character table.
| maximal subgroup of | #30627047, C5×#15691088, #30627430, … |
| maximal normal subgroup of | #30627047, C5×#15691088, #30627430, … |
| maximal quotient of | C5×#15691088, #30627052, #30627056, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3125, 31);