Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C67 ⋊ #1841 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #14524001 |
| commutator | C2×C4×C67 | C22×C4 |
| Frattini | C22×C4 | C22×D67 |
| Fitting | #339×C67 | C2 |
| socle | C2×C67 | #383 |
| hypercenter | #339 | D67 |
| hypocenter | C67 | #1841 |
| LCS | G ▷ C2×C4×C67 ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #339 | |
162 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 162 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=f,b^2=de,c^2=d,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,f^c=fg,h^c=h^66956 conjugacy classes — view the full character table.
| maximal subgroup of | #1841×D67, C2×#22560355, #30856098, … |
| maximal normal subgroup of | #1841×D67, C2×#22560355, #30856098, … |
| maximal quotient of | #30749548, #30749552, #30749556, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 4656);