Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C67 ⋊ #2567 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D67 |
| commutator, socle | C23×C67 | C24 |
| Fitting | C4×C4⋊C4×C67 | C2 |
| hypercenter | C4×C4⋊C4 | D67 |
| hypocenter | C67 | #2567 |
| LCS | G ▷ C23×C67 ▷ C67 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 | |
500 conjugacy classes of subgroups; only the 205 normal subgroups are shown.
This group has 500 conjugacy classes of subgroups — too many to draw. Only its 205 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=eg,b^2=eg,c^2=f,d^2=e,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 | #30753190, #30753230, #30753524, … |
| maximal normal subgroup of | #30753190, #30753230, #30753524, … |
| maximal quotient of | #30778107, #30778242, #30778704, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 9718);