Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C22×C4⋊C4×C67 ⋊ C2 |
| Sylow | C67 ⋊ #2591 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D67 |
| commutator, socle | C23×C67 | C24 |
| Fitting | C22×C4⋊C4×C67 | C2 |
| hypercenter | C22×C4⋊C4 | D67 |
| hypocenter | C67 | #2591 |
| LCS | G ▷ C23×C67 ▷ C67 | |
| UCS | C1 ◁ C23 ◁ C22×C4⋊C4 | |
596 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 596 conjugacy classes of subgroups — too many to draw. Only its 157 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=fg,b^2=eg,c^2=g,b^a=be,c^a=cf,d^a=de,h^a=h^66,c^b=cg1358 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30753279, #30753281, #30753299, … |
| maximal normal subgroup of | #30753279, #30753281, #30753299, … |
| maximal quotient of | #30778226, #30778591, #30779239, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 9990);