Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C64×C67 ⋊ C2 |
| Sylow | C67 ⋊ SD128 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D2144 |
| commutator | C32×C67 | C22 |
| Frattini | C32 | C2×D67 |
| Fitting | C64×C67 | C2 |
| socle | C2×C67 | D32 |
| hypercenter | C64 | D67 |
| hypocenter | C67 | SD128 |
| LCS | G ▷ C32×C67 ▷ C16×C67 ▷ C8×C67 ▷ C4×C67 ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ C64 | |
38 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,67|a^2=g,c^2=de,d^2=ef,e^2=fg,f^2=g,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,h^a=h^66,c^b=cd,d^b=de,e^b=ef,f^b=fg,h^b=h^662147 conjugacy classes. The character table is too large to display.
| maximal subgroup of | SD128×D67, C2×#22558425, #30776884, … |
| maximal normal subgroup of | SD128×D67, C2×#22558425, #30776884, … |
| maximal quotient of | #30743710, #30743713, #30743733, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 2726);