Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| Fitting | C2×C4⋊Q8×C67 ⋊ C2 |
| Sylow | C67 ⋊ #2858 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D67 |
| commutator, socle | C23×C67 | C24 |
| Fitting | C2×C4⋊Q8×C67 | C2 |
| hypercenter | C2×C4⋊Q8 | D67 |
| hypocenter | C67 | #2858 |
| LCS | G ▷ C23×C67 ▷ C67 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊Q8 | |
752 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 752 conjugacy classes of subgroups — too many to draw. Only its 197 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,67|b^2=g,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=dg,h^a=h^66,h^b=h^66,h^d=h^66968 conjugacy classes — view the full character table.
| maximal subgroup of | #30754880, #30754890, #30754895, … |
| maximal normal subgroup of | #30754880, #30754890, #30754895, … |
| maximal quotient of | #30787022, #30787100, #30787173, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 12868);