Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C67 ⋊ #2169 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C8 | C22×D67 |
| commutator, socle | C22×C67 | C22×C8 |
| Fitting | C8×D4×C67 | C2 |
| hypercenter | C8×D4 | D67 |
| hypocenter | C67 | #2169 |
| LCS | G ▷ C22×C67 ▷ C67 | |
| UCS | C1 ◁ C2×C8 ◁ C8×D4 | |
162 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 87 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=e,c^2=e,e^2=f,f^2=g,b^a=bd,c^a=cg,h^a=h^66,c^b=cg,h^c=h^661376 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30767542, #30767544, #30767545, … |
| maximal normal subgroup of | #30767542, #30767544, #30767545, … |
| maximal quotient of | #30767548, #30767619, #30767637, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 6342);