Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | #564×C67 ⋊ C2 |
| Sylow | C67 ⋊ #3556 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D67 |
| commutator, socle | C22×C67 | C25 |
| Fitting | #564×C67 | C2 |
| hypercenter | #564 | D67 |
| hypocenter | C67 | #3556 |
| LCS | G ▷ C22×C67 ▷ C67 | |
| UCS | C1 ◁ C22 ◁ #564 | |
890 conjugacy classes of subgroups; only the 455 normal subgroups are shown.
This group has 890 conjugacy classes of subgroups — too many to draw. Only its 455 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=fg,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,h^c=h^66,h^e=h^66665 conjugacy classes — view the full character table.
| maximal subgroup of | #30837729, #30837733, #30837762, … |
| maximal normal subgroup of | #30837729, #30837733, #30837762, … |
| maximal quotient of | #31018739, #31019473, #31021102, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 19004);