Finite Group DatabasebetaFinite Group Databasebeta| order | 15625 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C52×C25 · Schur multiplier (H2) = C5×C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C53 | C53 |
| commutator | C52 | C52×C25 |
| LCS | G ▷ C52 ▷ C1 | |
| UCS | C1 ◁ C53 ◁ G | |
| LECS | G ▷ C53 ▷ C1 | |
482 conjugacy classes of subgroups; only the 207 normal subgroups are shown.
This group has 482 conjugacy classes of subgroups — too many to draw. Only its 207 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
5^6|a^5=e^4,b^5=f,c^5=df,c^a=cd,c^b=ce1225 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15625, 121);