Finite Group DatabasebetaFinite Group Databasebeta| order | 1000 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C22⋊D125 |
| Fitting | C22×C125 ⋊ C2 |
| Sylow | C125 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D125 |
| commutator, cosocle | C2×C125 | C22 |
| Frattini | C2×C25 | C2×D5 |
| Fitting | C22×C125 | C2 |
| socle | C2×C5 | C2×D25 |
| hypercenter | C22 | D125 |
| hypocenter | C125 | D4 |
| LCS | G ▷ C2×C125 ▷ C125 | |
| UCS | C1 ◁ C2 ◁ C22 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 500 |
Polycyclic presentation:
2^3,5^3|d^5=e,e^5=f,b^a=bc,d^a=d^4e^4f^4,e^a=e^4f^4,f^a=f^4253 conjugacy classes — view the full character table.
| maximal subgroup of | #14522075, D4×D125, #14522077, … |
| maximal normal subgroup of | #14522075, D4×D125, #14522077, … |
| maximal quotient of | #14522049, #14522051, #14522052, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1000, 8);