Finite Group DatabasebetaFinite Group Databasebeta| order | 500 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽5) |
| automorphism | Aut(G) = C2×#92539 (order 1000) · Inn(G) = C52 · Out(G) = C2×AGL(1,𝔽5) |
| homology | Abelianization (H1) = C4×C52 · Schur multiplier (H2) = C1 |
| direct product | C4×5−1+2 |
| unambiguous semidirect product | (C4×C25)⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C52 |
| commutator | C5 | C4×C52 |
| Frattini, socle, cosocle | C2×C5 | C2×C52 |
| prime radical | C2×C52 | C2×C5 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ G | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 100 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2,4,8,15,3,6,10,17,22,7,13,19,16,23,14,12,9,18,24,21,5,11,20,25)(26,27,28,29);(2,5,12,13,6)(4,9,10,11,19)(8,16,20,17,18)(15,22,23,24,25)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(500, 14);