Finite Group DatabasebetaFinite Group Databasebeta| order | 500 |
| isoclinism | isoclinic class of D5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C52 · Schur multiplier (H2) = C2×C5 |
| direct product | C2×C52×D5 |
| unambiguous semidirect product | C5⋊(C22×C52) |
| unambiguous semidirect product | (C2×C5)⋊(C2×C52) |
| Fitting, socle | C2×C53 ⋊ C2 |
| derived | C5 ⋊ C22×C52 |
| Sylow | C53 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C52 | D5 |
| commutator, cosocle, hypocenter | C5 | C22×C52 |
| Frattini | C1 | G |
| Fitting, socle | C2×C53 | C2 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2×C52 | |
128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 100 |
A minimal faithful permutation representation — 2 generators on 17 points:
(4,5)(6,7)(8,9,10,11,12);(1,2)(3,4,6,7,5)(13,14,15,16,17)200 conjugacy classes — view the full character table.
| maximal subgroup of | C5×D5×Dic5, C5×#6037, C5×#6038, … |
| maximal normal subgroup of | C5×D5×Dic5, C5×#6037, C5×#6038, … |
| maximal quotient of | C52×Dic10, C4×C52×D5, C52×D20, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(500, 53);