Finite Group DatabasebetaFinite Group Databasebeta| order | 2000 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | Q8×C125 ⋊ C2 |
| Sylow | C125 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×D125 |
| commutator, cosocle | C2×C125 | C23 |
| Frattini | C2×C25 | C22×D5 |
| Fitting | Q8×C125 | C2 |
| socle | C2×C5 | C22×D25 |
| hypercenter | Q8 | D125 |
| hypocenter | C125 | #41 |
| LCS | G ▷ C2×C125 ▷ C125 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
80 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1000 |
Polycyclic presentation:
2^4,5^3|b^2=d,c^2=d,e^5=f,f^5=g,c^a=cd,e^a=e^4f^4g^4,f^a=f^4g^4,g^a=g^4,c^b=cd320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2000, 42);