Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | #344×C67 ⋊ C2 |
| Sylow | C67 ⋊ #2008 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #16863306 |
| commutator | C22×C4×C67 | C23 |
| Frattini | C22×C4 | C22×D67 |
| Fitting | #344×C67 | C2 |
| socle | C2×C67 | #392 |
| hypercenter | #344 | D67 |
| hypocenter | C67 | #2008 |
| LCS | G ▷ C22×C4×C67 ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #344 | |
380 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,67|d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,h^a=h^66,c^b=cf,d^b=dg,h^b=h^66,d^c=dg,e^c=eg,h^c=h^66749 conjugacy classes — view the full character table.
| maximal subgroup of | #2008×D67, C2×#22561265, #30882838, … |
| maximal normal subgroup of | #2008×D67, C2×#22561265, #30882838, … |
| maximal quotient of | #30756142, #30756156, #30756170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 5566);