Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C128 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C67⋊C128 |
| derived, Sylow | C67 ⋊ C128 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C64 | D67 |
| commutator, hypocenter | C67 | C128 |
| Fitting | C64×C67 | C2 |
| socle | C2×C67 | C64 |
| LCS | G ▷ C67 | |
| UCS | C1 ◁ C64 | |
16 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,67|a^2=b,b^2=c,c^2=d,d^2=e,e^2=f,f^2=g,h^a=h^662240 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C128×D67, C2×C67⋊C128, #30743741, … |
| maximal normal subgroup of | C128×D67, C2×C67⋊C128, #30743741, … |
| maximal quotient of | C67⋊C256, C2×C67⋊C128 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 2329);