Finite Group DatabasebetaFinite Group Databasebeta| order | 6250 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, Sylow | #15691062 ⋊ C2 |
| derived | C52⋊C25 ⋊ C2×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | #11929380 |
| commutator, hypocenter | C52⋊C25 | C2×C5 |
| Frattini | C53 | C5×D5 |
| Fitting | #15691062 | C2 |
| socle | C52 | (C5⋊D5)⋊C5 |
| LCS | G ▷ C52⋊C25 | |
| UCS | C1 ◁ C5 | |
72 conjugacy classes of subgroups.
Polycyclic presentation:
2,5^5|c^5=e,c^a=c^4e^4,d^a=d^4f,e^a=e^4,c^b=cd,d^b=de,d^c=df112 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6250, 84);