Finite Group DatabasebetaFinite Group Databasebeta| order | 3125 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C54 · Schur multiplier (H2) = C55 |
| extraspecial | 5−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C5 | C54 |
| prime radical | C5×H(3,𝔽5) | C5 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C5 ◁ G | |
| LECS | G ▷ C5 ▷ C1 | |
1158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 125 |
Polycyclic presentation:
5^5|a^5=e,b^a=be,d^a=de,c^b=ce629 conjugacy classes — view the full character table.
| Fitting subgroup of | #72229811 |
| maximal subgroup of | #30626978, C5×5−1+4, #30627405, … |
| maximal normal subgroup of | #30626978, C5×5−1+4, #30627405, … |
| maximal quotient of | C5×5−1+4, #30626982, #30626983, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3125, 76);