Finite Group DatabasebetaFinite Group Databasebeta| order | 2000 |
| isoclinism | isoclinic class of #92534 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D25⋊#168 |
| derived, Sylow | C5×C25 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #92534 |
| commutator, hypocenter | C5×C25 | C2×C8 |
| Frattini | C2×C5 | C2×#1042 |
| Fitting | C4×C5×C25 | C4 |
| socle | C2×C52 | C2×AGL(1,𝔽5) |
| prime radical | C2×C5×D25 | C4 |
| cosocle | C5×Dic25 | C22 |
| LCS | G ▷ C5×C25 | |
| UCS | C1 ◁ C4 | |
72 conjugacy classes of subgroups.
| transitive degree | μt(G) = 400 |
Polycyclic presentation:
2^4,5^3|a^2=c,b^2=d,c^2=d,e^5=g,e^a=e^2g,f^a=f^4,g^a=g^2,e^c=e^4g^4,g^c=g^4152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2000, 304);