Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | D4×Q8×C67 ⋊ C2 |
| Sylow | C67 ⋊ #3499 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D67 |
| commutator, socle | C22×C67 | C25 |
| Fitting | D4×Q8×C67 | C2 |
| hypercenter | D4×Q8 | D67 |
| hypocenter | C67 | #3499 |
| LCS | G ▷ C22×C67 ▷ C67 | |
| UCS | C1 ◁ C22 ◁ D4×Q8 | |
1476 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,67|a^2=f,c^2=g,e^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df,h^b=h^66,h^d=h^66,h^e=h^66869 conjugacy classes — view the full character table.
| maximal subgroup of | #30832494, #30832500, #30832582, … |
| maximal normal subgroup of | #30832494, #30832500, #30832582, … |
| maximal quotient of | #31015934, #31016201, #31016220, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 18294);