Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Sylow | C67 ⋊ #3502 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D67 |
| commutator, socle | C22×C67 | C25 |
| Fitting | C2×C4×Q8×C67 | C2 |
| hypercenter | C2×C4×Q8 | D67 |
| hypocenter | C67 | #3502 |
| LCS | G ▷ C22×C67 ▷ C67 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 | |
964 conjugacy classes of subgroups; only the 553 normal subgroups are shown.
This group has 964 conjugacy classes of subgroups — too many to draw. Only its 553 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=f,c^2=g,d^2=f,b^a=bf,c^a=cg,e^a=ef,h^a=h^66,d^b=df1364 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30747393, #30747396, #30747830, … |
| maximal normal subgroup of | #30747393, #30747396, #30747830, … |
| maximal quotient of | #31015928, #31016271, #31016769, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 18320);