Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C67 ⋊ #1305 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C67⋊#95 |
| commutator | C23×C67 | C42 |
| Frattini | C23×C4 | C2×D67 |
| Fitting | #380×C67 | C2 |
| socle | C22×C67 | #95 |
| hypercenter | #380 | D67 |
| hypocenter | C67 | #1305 |
| LCS | G ▷ C23×C67 ▷ C22×C67 ▷ C67 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #380 | |
200 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 61 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=d,b^2=e,d^2=f,b^a=bc,c^a=cf,e^a=eg,h^a=h^66,c^b=cg,d^b=df950 conjugacy classes — view the full character table.
| maximal subgroup of | #1305×D67, C2×#22558115, #30746163, … |
| maximal normal subgroup of | #1305×D67, C2×#22558115, #30746163, … |
| maximal quotient of | #30742402, #30742405, #30742406, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 2416);