Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C2×2−1+4×C67 ⋊ C2 |
| Sylow | C67 ⋊ #3500 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D67 |
| commutator, socle | C22×C67 | C25 |
| Fitting | C2×2−1+4×C67 | C2 |
| hypercenter | C2×2−1+4 | D67 |
| hypocenter | C67 | #3500 |
| LCS | G ▷ C22×C67 ▷ C67 | |
| UCS | C1 ◁ C22 ◁ C2×2−1+4 | |
1540 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,67|c^2=fg,e^2=f,b^a=bf,c^a=cg,e^a=ef,h^a=h^66,d^b=df,h^c=h^661166 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30771396, #30865035, #30865042, … |
| maximal normal subgroup of | #30771396, #30865035, #30865042, … |
| maximal quotient of | #31015938, #31016213, #31016241, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 18304);