Finite Group DatabasebetaFinite Group Databasebeta| order | 500 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) = C2×C4×C5×AGL(1,𝔽5) (order 800) · Inn(G) = D5 · Out(G) = C22×C4×C5 |
| homology | Abelianization (H1) = C22×C25 · Schur multiplier (H2) = C2 |
| direct product | C2×D5×C25 |
| unambiguous semidirect product | C5⋊(C22×C25) |
| unambiguous semidirect product | (C2×C5)⋊(C2×C25) |
| Fitting | C2×C5×C25 ⋊ C2 |
| derived | C5 ⋊ C22×C25 |
| Sylow | C5×C25 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C25 | D5 |
| commutator, hypocenter | C5 | C22×C25 |
| Frattini | C5 | C2×C5×D5 |
| Fitting | C2×C5×C25 | C2 |
| socle | C2×C52 | C2×C5 |
| prime radical | C2×C5×D5 | C5 |
| cosocle | C52 | C22×C5 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2×C25 | |
38 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 100 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(4,5)(6,7)(8,12,20,28,32,10,16,25,31,18,13,21,29,14,23,17,26,11,19,27,22,9,15,24,30);(1,2)(3,4,6,7,5)200 conjugacy classes — view the full character table.
| maximal subgroup of | D5×Dic25, #11758655, #11758656, … |
| maximal normal subgroup of | D5×Dic25, #11758655, #11758656, … |
| maximal quotient of | C25×Dic10, C4×D5×C25, C25×D20, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(500, 29);