Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C67 ⋊ C4×C4⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D67 |
| commutator | C22×C67 | C23×C4 |
| Fitting | #391×C67 | C2 |
| socle | C23×C67 | C24 |
| hypercenter | #391 | D67 |
| hypocenter | C67 | C4×C4⋊Q8 |
| LCS | G ▷ C22×C67 ▷ C67 | |
| UCS | C1 ◁ C23 ◁ #391 | |
588 conjugacy classes of subgroups; only the 237 normal subgroups are shown.
This group has 588 conjugacy classes of subgroups — too many to draw. Only its 237 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=g,b^2=e,c^2=f,d^2=eg,b^a=be,c^a=cf,h^c=h^66,h^d=h^66980 conjugacy classes — view the full character table.
| maximal subgroup of | #30743811, #30743898, #30743957, … |
| maximal normal subgroup of | #30743811, #30743898, #30743957, … |
| maximal quotient of | #30777023, #30777029, #391×Dic67, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 7079);