Finite Group DatabasebetaFinite Group Databasebeta| order | 525 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C52 · Schur multiplier (H2) = C5 |
| direct product | C52×ΓL(1,𝔽8) |
| unambiguous semidirect product | C7⋊(C3×C52) |
| unambiguous semidirect product | (C5×C7)⋊(C3×C5) |
| Fitting, socle, Hall | C52×C7 ⋊ C3 |
| derived, Sylow | C7 ⋊ C3×C52 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C52 | ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C7 | C3×C52 |
| Frattini | C1 | G |
| Fitting, socle | C52×C7 | C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C52 | |
32 conjugacy classes of subgroups.
| transitive degree | μt(G) = 175 |
125 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(525, 3);