Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | #504×C67 ⋊ C2 |
| Sylow | C67 ⋊ #2171 |
| subgroup | G/H | |
|---|---|---|
| center | C8 | C2×C22⋊D67 |
| commutator | C4×C67 | C22×C8 |
| Frattini | C2×C8 | C22×D67 |
| Fitting | #504×C67 | C2 |
| socle | C2×C67 | C8×D4 |
| hypercenter | #504 | D67 |
| hypocenter | C67 | #2171 |
| LCS | G ▷ C4×C67 ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C8 ◁ C2×C16 ◁ #504 | |
144 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 65 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,d^2=g,e^2=f,f^2=g,b^a=bd,c^a=cg,d^a=dg,d^b=dg,h^b=h^661364 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #2171×D67, C2×#22562049, #31009069, … |
| maximal normal subgroup of | #2171×D67, C2×#22562049, #31009069, … |
| maximal quotient of | #30767539, #30767541, #30767602, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 6350);