Finite Group DatabasebetaFinite Group Databasebeta| order | 250 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽5) |
| automorphism | Aut(G) = #92539 (order 500) · Inn(G) = C52 · Out(G) = AGL(1,𝔽5) |
| homology | Abelianization (H1) = C2×C52 · Schur multiplier (H2) = C1 |
| direct product | C2×5−1+2 |
| unambiguous semidirect product | (C2×C25)⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C52 |
| commutator, Frattini, cosocle | C5 | C2×C52 |
| prime radical | C2×C52 | C5 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ G | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 50 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2)(3,4,6,10,17,5,8,12,19,24,9,15,21,18,25,16,14,11,20,26,23,7,13,22,27);(4,7,14,15,8)(6,11,12,13,21)(10,18,22,19,20)(17,24,25,26,27)58 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(250, 11);