Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| Fitting | #585×C67 ⋊ C2 |
| Sylow | C67 ⋊ H(7,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C25×D67 |
| commutator, socle | C2×C67 | C26 |
| Fitting | #585×C67 | C2 |
| hypercenter | #585 | D67 |
| hypocenter | C67 | H(7,𝔽2) |
| LCS | G ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C2 ◁ #585 | |
5992 conjugacy classes of subgroups; only the 3201 normal subgroups are shown.
This group has 5992 conjugacy classes of subgroups — too many to draw. Only its 3201 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,67|b^a=bg,f^a=fg,e^b=eg,d^c=dg,h^e=h^66,h^f=h^661187 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #31004572, #31004574, #31004576, … |
| maximal normal subgroup of | #31004572, #31004574, #31004576, … |
| maximal quotient of | #31762447, #31762449, #31762456, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 19321);