Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C67 ⋊ #1916 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22×C4 | C22×D67 |
| commutator | C22×C67 | C22×C8 |
| Fitting | C2×C4×C8×C67 | C2 |
| socle | C23×C67 | C22×C4 |
| hypercenter | C2×C4×C8 | D67 |
| hypocenter | C67 | #1916 |
| LCS | G ▷ C22×C67 ▷ C67 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C8 | |
240 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 109 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=f,c^2=d,f^2=g,b^a=bd,c^a=ce,h^a=h^662168 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #1916×D67, C2×#22560766, #30857885, … |
| maximal normal subgroup of | #1916×D67, C2×#22560766, #30857885, … |
| maximal quotient of | #30753948, #30753955, #30753962, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 5067);