Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C67 · Schur multiplier (H2) = C22×C4 |
| direct product | C67×#1975 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C67 | C2×D4 |
| commutator | C2×C4 | C22×C4×C67 |
| Frattini, cosocle | C22×C4 | C23×C67 |
| socle | C22×C67 | C4×D4 |
| prime radical | C2×#135×C67 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C67 ◁ C22×C8×C67 ◁ G | |
292 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
2144 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 707);