Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | C7×C27 ⋊ D4 |
| Sylow | C27 ⋊ C22⋊D7 |
| Sylow | C7 ⋊ #6171 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D7×D27 |
| commutator, Fitting, cosocle | C2×C7×C27 | C22 |
| Frattini | C2×C9 | D3×D7 |
| socle | C2×C3×C7 | C2×D9 |
| hypocenter | C7×C27 | D4 |
| LCS | G ▷ C2×C7×C27 ▷ C7×C27 | |
| UCS | C1 ◁ C2 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 756 |
Polycyclic presentation:
2^3,3,7,3^2|d^3=f,f^3=g,b^a=bc,e^a=e^6,d^b=d^2f^2g^2,f^b=f^2g^2,g^b=g^2147 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 17);