Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C8×Q8×C67 ⋊ C2 |
| Sylow | C67 ⋊ #2988 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D67 |
| commutator | C2×C67 | C23×C8 |
| Fitting | C8×Q8×C67 | C2 |
| socle | C22×C67 | C23×C4 |
| hypercenter | C8×Q8 | D67 |
| hypocenter | C67 | #2988 |
| LCS | G ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C2×C4 ◁ C8×Q8 | |
432 conjugacy classes of subgroups; only the 217 normal subgroups are shown.
This group has 432 conjugacy classes of subgroups — too many to draw. Only its 217 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,d^2=e,f^2=g,b^a=be,d^a=de,c^b=ce,h^b=h^66,h^c=h^66,h^d=h^661388 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30825264, #30825268, #30825275, … |
| maximal normal subgroup of | #30825264, #30825268, #30825275, … |
| maximal quotient of | #30854617, #30854718, #30854869, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 13620);