Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C67 ⋊ #2971 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D67 |
| commutator, socle | C22×C67 | C23×C4 |
| Fitting | C4×#41×C67 | C2 |
| hypercenter | C4×#41 | D67 |
| hypocenter | C67 | #2971 |
| LCS | G ▷ C22×C67 ▷ C67 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#41 | |
402 conjugacy classes of subgroups; only the 221 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 221 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,d^2=e,f^2=g,b^a=be,c^a=cg,d^a=dg,h^a=h^66,c^b=cg,d^c=dg,h^c=h^661370 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30751631, #30751635, #30751645, … |
| maximal normal subgroup of | #30751631, #30751635, #30751645, … |
| maximal quotient of | #30854121, #30854272, Q8×#14523954, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 13530);