Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | #585×C67 ⋊ C2 |
| Sylow | C67 ⋊ #3018 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #14524012 |
| commutator | C2×C4×C67 | C24 |
| Frattini | C2×C4 | C23×D67 |
| Fitting | #585×C67 | C2 |
| socle | C2×C67 | C2×#120 |
| hypercenter | #585 | D67 |
| hypocenter | C67 | #3018 |
| LCS | G ▷ C2×C4×C67 ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #585 | |
724 conjugacy classes of subgroups; only the 161 normal subgroups are shown.
This group has 724 conjugacy classes of subgroups — too many to draw. Only its 161 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,67|d^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=h^66,e^b=eg,f^b=fg,e^c=eg1154 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30773150, #30773155, #30773160, … |
| maximal normal subgroup of | #30773150, #30773155, #30773160, … |
| maximal quotient of | #30827937, #30827988, #30828039, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 13859);