Finite Group DatabasebetaFinite Group Databasebeta| order | 500 |
| isoclinism | isoclinic class of D25⋊C5 |
| automorphism | Aut(G) = C2×AGL(1,ℤ/25) (order 1000) · Inn(G) = D25⋊C5 · Out(G) = C22 |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2 |
| direct product | C2×D25⋊C5 |
| unambiguous semidirect product | D25⋊(C2×C5) |
| unambiguous semidirect product | (C2×D25)⋊C5 |
| unambiguous semidirect product | 5−1+2⋊C22 |
| unambiguous semidirect product | (C2×5−1+2)⋊C2 |
| Fitting | C2×5−1+2 ⋊ C2 |
| derived | C25 ⋊ C22×C5 |
| Sylow | 5−1+2 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D25⋊C5 |
| commutator, cosocle, hypocenter | C25 | C22×C5 |
| Frattini | C5 | C2×C5×D5 |
| Fitting | C2×5−1+2 | C2 |
| socle | C2×C5 | C5×D5 |
| LCS | G ▷ C25 | |
| UCS | C1 ◁ C2 | |
34 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 50 |
A minimal faithful permutation representation — 2 generators on 27 points:
(4,11,13,21,18,6,7,20,22,12)(5,9)(8,24,26,17,23,15,16,25,14,27)(10,19);(1,2)(3,18,23,25,6,9,22,16,15,11,19,13,14,17,20,10,7,8,24,21,5,4,26,27,12)44 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D25⋊C5, C4⋊(D25⋊C5), C22⋊(D25⋊C5), … |
| maximal normal subgroup of | C4×D25⋊C5, C4⋊(D25⋊C5), C22⋊(D25⋊C5), … |
| maximal quotient of | 5−1+2⋊Q8, C4×D25⋊C5, C4⋊(D25⋊C5), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(500, 31);