Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C67 ⋊ #1375 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×Dic67 |
| commutator | C4×C67 | C4×C8 |
| Frattini | C22×C8 | C2×D67 |
| Fitting | C2×M32×C67 | C2 |
| socle | C22×C67 | #97 |
| hypercenter | C2×M32 | D67 |
| hypocenter | C67 | #1375 |
| LCS | G ▷ C4×C67 ▷ C67 | |
| UCS | C1 ◁ C2×C4 ◁ C2×M32 | |
140 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 77 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=d,b^2=e,c^2=g,d^2=f,f^2=g,b^a=bc,e^a=eg,d^b=dg,h^b=h^661364 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #1375×D67, C2×#22558288, #30770679, … |
| maximal normal subgroup of | #1375×D67, C2×#22558288, #30770679, … |
| maximal quotient of | #30742660, #30742672, #30742679, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 2589);