Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C25 ⋊ #559 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×D25 |
| commutator, cosocle | C22×C25 | C24 |
| Frattini, socle | C22×C5 | C23×D5 |
| Fitting | C25×#127 | C2 |
| prime radical | C5×#127 | D5 |
| hypercenter | #127 | D25 |
| hypocenter | C25 | #559 |
| LCS | G ▷ C22×C25 ▷ C25 | |
| UCS | C1 ◁ C22 ◁ #127 | |
354 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 800 |
Polycyclic presentation:
2^6,5^2|a^2=f,b^2=f,c^2=e,g^5=h,b^a=bf,d^a=de,g^a=g^4h^4,h^a=h^4,d^b=df,d^c=de193 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 1389);