Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | Q82×C67 ⋊ C2 |
| Sylow | C67 ⋊ #3385 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×C22⋊D67 |
| commutator | C2×C4×C67 | C24 |
| Frattini | C2×C4 | C23×D67 |
| Fitting | Q82×C67 | C2 |
| socle | C22×C67 | C22×D4 |
| hypercenter | Q82 | D67 |
| hypocenter | C67 | #3385 |
| LCS | G ▷ C2×C4×C67 ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 ◁ Q82 | |
340 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 340 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,67|a^2=eg,b^2=e,d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,c^b=cg,d^b=de,f^c=fg,h^c=h^66854 conjugacy classes — view the full character table.
| maximal subgroup of | #30845329, #30845331, #30845374, … |
| maximal normal subgroup of | #30845329, #30845331, #30845374, … |
| maximal quotient of | #30872452, #30872489, #30874121, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 17416);