Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C67 ⋊ #2070 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #16863320 |
| commutator | C22×C4×C67 | C23 |
| Frattini | C22×C4 | C22×D67 |
| Fitting | #338×C67 | C2 |
| socle | C2×C67 | #396 |
| hypercenter | #338 | D67 |
| hypocenter | C67 | #2070 |
| LCS | G ▷ C22×C4×C67 ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #338 | |
196 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,67|a^2=de,b^2=d,c^2=g,d^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,h^a=h^66,c^b=cf,f^b=fg,h^b=h^66,d^c=dg,h^c=h^66749 conjugacy classes — view the full character table.
| maximal subgroup of | #2070×D67, C2×#22561611, #30884554, … |
| maximal normal subgroup of | #2070×D67, C2×#22561611, #30884554, … |
| maximal quotient of | #30760468, #30760482, #30760496, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 5912);