Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | Dic67⋊C32 |
| Sylow | C67 ⋊ C4⋊C32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C16 | C2×D67 |
| commutator | C2×C67 | C2×C32 |
| Fitting | C2×C32×C67 | C2 |
| socle | C22×C67 | C2×C16 |
| hypercenter | C2×C32 | D67 |
| hypocenter | C67 | C4⋊C32 |
| LCS | G ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C2×C16 ◁ C2×C32 | |
62 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,67|a^2=d,b^2=c,d^2=e,e^2=f,f^2=g,b^a=bc,h^b=h^662192 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C4⋊C32×D67, C2×Dic67⋊C32, #30776312, … |
| maximal normal subgroup of | C4⋊C32×D67, C2×Dic67⋊C32, #30776312, … |
| maximal quotient of | #30743496, #30743503, #30743517, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 2705);