Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | #475×C67 ⋊ C2 |
| Sylow | C67 ⋊ #3242 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D67 |
| commutator | C2×C4×C67 | C24 |
| Frattini | C2×C4 | C23×D67 |
| Fitting | #475×C67 | C2 |
| socle | C22×C67 | C22×D4 |
| hypercenter | #475 | D67 |
| hypocenter | C67 | #3242 |
| LCS | G ▷ C2×C4×C67 ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #475 | |
308 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,67|a^2=g,b^2=e,c^2=e,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,h^a=h^66,c^b=ce,d^b=dg,h^b=h^66,f^c=fg,h^c=h^66,h^d=h^66653 conjugacy classes — view the full character table.
| maximal subgroup of | #3242×D67, C2×#22571484, #31728082, … |
| maximal normal subgroup of | #3242×D67, C2×#22571484, #31728082, … |
| maximal quotient of | #30865389, #30865626, #30866439, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 15785);