Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | Q64×C67 ⋊ C2 |
| Sylow | C67 ⋊ #2264 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D16×D67 |
| commutator | C16×C67 | C23 |
| Frattini | C16 | C22×D67 |
| Fitting | Q64×C67 | C2 |
| socle | C2×C67 | C2×D16 |
| hypercenter | Q64 | D67 |
| hypocenter | C67 | #2264 |
| LCS | G ▷ C16×C67 ▷ C8×C67 ▷ C4×C67 ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ Q64 | |
104 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,67|b^2=g,d^2=ef,e^2=fg,f^2=g,b^a=bd,c^a=cg,d^a=de,e^a=ef,f^a=fg,h^a=h^66,d^b=de,e^b=ef,f^b=fg,h^c=h^66653 conjugacy classes — view the full character table.
| maximal subgroup of | #2264×D67, C2×#22562528, #31013974, … |
| maximal normal subgroup of | #2264×D67, C2×#22562528, #31013974, … |
| maximal quotient of | #30776269, #30776276, #30776335, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 6829);