Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| isoclinism | isoclinic class of ΓL(1,𝔽64) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived | C7×C9 ⋊ C3×C8 |
| Hall | #4428 ⋊ C8 |
| Sylow | C7 ⋊ 3−1+2⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | ΓL(1,𝔽64) |
| commutator, hypocenter | C7×C9 | C3×C8 |
| Frattini | C3×C4 | D3×ΓL(1,𝔽8) |
| Fitting, cosocle | C4×C7×C9 | C2×C3 |
| socle | C2×C3×C7 | C3×Dic3 |
| prime radical | C2×C3×ΓL(1,𝔽8) | Dic3 |
| LCS | G ▷ C7×C9 | |
| UCS | C1 ◁ C4 | |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 504 |
Polycyclic presentation:
2,3,2^2,3,7,3|a^2=c,c^2=d,e^3=g,e^a=e^2g^2,g^a=g^2,e^b=eg^2,f^b=f^288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 49);