Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C67 ⋊ #2956 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×D4×D67 |
| commutator | C4×C67 | C23×C4 |
| Frattini | C2×C4 | C23×D67 |
| Fitting | #575×C67 | C2 |
| socle | C2×C67 | C2×C4×D4 |
| hypercenter | #575 | D67 |
| hypocenter | C67 | #2956 |
| LCS | G ▷ C4×C67 ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #575 | |
458 conjugacy classes of subgroups; only the 217 normal subgroups are shown.
This group has 458 conjugacy classes of subgroups — too many to draw. Only its 217 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=f,c^2=f,e^2=g,f^2=g,b^a=be,e^a=eg,h^a=h^66,e^b=eg,d^c=dg,h^c=h^66770 conjugacy classes — view the full character table.
| maximal subgroup of | #30876697, #30876701, #30876706, … |
| maximal normal subgroup of | #30876697, #30876701, #30876706, … |
| maximal quotient of | #30825270, #30825281, #30825461, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 13420);