Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C67 ⋊ #2108 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | C2×D67 |
| commutator | C2×C67 | C2×C4×C8 |
| Frattini | C2×C8 | C22×D67 |
| Fitting | C22×C16×C67 | C2 |
| socle | C22×C67 | C22×C8 |
| hypercenter | C22×C16 | D67 |
| hypocenter | C67 | #2108 |
| LCS | G ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C2×C16 ◁ C22×C16 | |
184 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 125 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,f^2=g,c^b=cg,h^b=h^662192 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30743525, #30743527, #30743528, … |
| maximal normal subgroup of | #30743525, #30743527, #30743528, … |
| maximal quotient of | #30764812, #30764821, C16×#11761612, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 6106);