Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | ΓL(1,𝔽8)×#625 |
| unambiguous semidirect product | (C7×#625)⋊C3 |
| Hall | C32×ΓL(1,𝔽8) ⋊ D4 |
| Sylow | C7 ⋊ C3×#625 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D32×ΓL(1,𝔽8) |
| commutator, Fitting, socle, cosocle | C2×C32×C7 | C22×C3 |
| hypocenter | C32×C7 | C3×D4 |
| LCS | G ▷ C2×C32×C7 ▷ C32×C7 | |
| UCS | C1 ◁ C2 | |
188 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
75 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#3802, AGL(1,𝔽7)×#625, #72246948, … |
| maximal normal subgroup of | ΓL(1,𝔽8)×#3802, AGL(1,𝔽7)×#625, #72246948, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 617);