Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C67 ⋊ #1771 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22×C4 | C22×D67 |
| commutator | C22×C67 | C2×C42 |
| Fitting | C4×C4⋊C4×C67 | C2 |
| socle | C23×C67 | C22×C4 |
| hypercenter | C4×C4⋊C4 | D67 |
| hypocenter | C67 | #1771 |
| LCS | G ▷ C22×C67 ▷ C67 | |
| UCS | C1 ◁ C22×C4 ◁ C4×C4⋊C4 | |
272 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 141 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=e,b^2=f,c^2=de,e^2=g,b^a=bd,c^a=cg,h^a=h^66,h^c=h^661376 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #1771×D67, C2×#22560084, #30853777, … |
| maximal normal subgroup of | #1771×D67, C2×#22560084, #30853777, … |
| maximal quotient of | #30746967, #30746973, #30746980, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 4385);