Finite Group DatabasebetaFinite Group Databasebeta| order | 500 |
| isoclinism | isoclinic class of AGL(1,ℤ/25) |
| automorphism | Aut(G) = AGL(1,ℤ/25) (order 500) · Inn(G) = G · Out(G) = C1 |
| homology | Abelianization (H1) = C4×C5 · Schur multiplier (H2) = C1 |
| affine general linear | AGL(1,ℤ/25) |
| unambiguous semidirect product | #1035⋊C5 |
| Fitting, Sylow | 5−1+2 ⋊ C4 |
| derived | C25 ⋊ C4×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | C25 | C4×C5 |
| Frattini, socle | C5 | C5×AGL(1,𝔽5) |
| Fitting | 5−1+2 | C4 |
| prime radical | D25⋊C5 | C2 |
| cosocle | D25 | C2×C5 |
| LCS | G ▷ C25 | |
| UCS | C1 | |
19 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 25 |
A minimal faithful permutation representation — 2 generators on 25 points:
(2,10,24,18,21,4,14,23,25,17,6,19,13,7,15,11,5,22,16,12)(3,8,9,20);(1,2,7,19,14,9,21,4,12,24,20,25,22,23,16,8,13,17,11,15,3,5,10,18,6)Polycyclic presentation:
2,5,2,5^2|a^2=c,d^5=e^4,d^a=d^2e^4,e^a=e^2,d^b=de,d^c=d^4e,e^c=e^426 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(500, 18);