Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #416×C67 ⋊ C2 |
| Sylow | C67 ⋊ #2053 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #14524029 |
| commutator | C22×C4×C67 | C23 |
| Frattini | C22×C4 | C22×D67 |
| Fitting | #416×C67 | C2 |
| socle | C2×C67 | #394 |
| hypercenter | #416 | D67 |
| hypocenter | C67 | #2053 |
| LCS | G ▷ C22×C4×C67 ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #416 | |
264 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,67|c^2=d,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=cf,d^b=dg,f^b=fg,h^b=h^66950 conjugacy classes — view the full character table.
| maximal subgroup of | #2053×D67, C2×#22561513, #30883750, … |
| maximal normal subgroup of | #2053×D67, C2×#22561513, #30883750, … |
| maximal quotient of | #30759563, #30759565, #30759570, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 5814);