Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C67 ⋊ #1917 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22×C4 | C22×D67 |
| commutator, socle | C23×C67 | C22×C4 |
| Fitting | C4×C4⋊C4×C67 | C2 |
| hypercenter | C4×C4⋊C4 | D67 |
| hypocenter | C67 | #1917 |
| LCS | G ▷ C23×C67 ▷ C67 | |
| UCS | C1 ◁ C22×C4 ◁ C4×C4⋊C4 | |
228 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 105 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=f,c^2=d,f^2=g,b^a=bd,c^a=ce,h^a=h^66,c^b=cg,h^b=h^661364 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #1917×D67, C2×#22560775, #30857893, … |
| maximal normal subgroup of | #1917×D67, C2×#22560775, #30857893, … |
| maximal quotient of | #30753978, #30753985, #30753992, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 5076);