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 ⋊ #3362 |
| 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 | #3362 |
| LCS | G ▷ C2×C4×C67 ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 ◁ Q82 | |
392 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 141 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=e,c^2=g,d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,h^a=h^66,d^b=de,f^c=fg860 conjugacy classes — view the full character table.
| maximal subgroup of | #30845367, #30845376, #30845397, … |
| maximal normal subgroup of | #30845367, #30845376, #30845397, … |
| maximal quotient of | Q8×#14523967, #30872616, #30874767, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 17181);