Finite Group DatabasebetaFinite Group Databasebeta| order | 510 |
| isoclinism | isoclinic class of D17 |
| automorphism | Aut(G) = C2×C4×AGL(1,𝔽17) (order 2176) · Inn(G) = D17 · Out(G) = C2×C4×C8 |
| homology | Abelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1 |
| direct product | C3×C5×D17 |
| unambiguous semidirect product | C17⋊(C2×C3×C5) |
| Fitting, socle, Hall | C3×C5×C17 ⋊ C2 |
| derived, Sylow | C17 ⋊ C2×C3×C5 |
| Hall | C5×C17 ⋊ C2×C3 |
| Hall | C3×C17 ⋊ C2×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C5 | D17 |
| commutator, cosocle, hypocenter | C17 | C2×C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C3×C5×C17 | C2 |
| LCS | G ▷ C17 | |
| UCS | C1 ◁ C3×C5 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 255 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,3)(4,5,6,7,8)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25);(9,10,12,14,16,18,20,22,24,25,23,21,19,17,15,13,11)150 conjugacy classes — view the full character table.
| maximal subgroup of | C3×C5×#595, C3×#66772, C5×C3⋊#595, … |
| maximal normal subgroup of | C3×C5×#595, C3×#66772, C5×C3⋊#595, … |
| maximal quotient of | C3×C5×Dic17, C2×C3×C5×D17, C5×C9×D17, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(510, 1);