Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | Q8⋊C8×C67 ⋊ C2 |
| Sylow | C67 ⋊ #1649 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #14524013 |
| commutator | C42×C67 | C23 |
| Frattini | C42 | C22×D67 |
| Fitting | Q8⋊C8×C67 | C2 |
| socle | C22×C67 | #120 |
| hypercenter | Q8⋊C8 | D67 |
| hypocenter | C67 | #1649 |
| LCS | G ▷ C42×C67 ▷ C22×C67 ▷ C67 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ Q8⋊C8 | |
234 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 59 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=g,b^2=g,c^2=g,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,h^a=h^66,d^b=df,e^b=eg,h^b=h^66,d^c=dg,e^c=eg950 conjugacy classes — view the full character table.
| maximal subgroup of | #1649×D67, C2×#22559460, #30831669, … |
| maximal normal subgroup of | #1649×D67, C2×#22559460, #30831669, … |
| maximal quotient of | #30744654, #30744660, #30744695, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 3761);