Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| Sylow | C67 ⋊ #1726 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D67 |
| commutator | C2×C67 | C43 |
| Frattini | C22×C4 | C22×D67 |
| Fitting | C2×C4×C8×C67 | C2 |
| socle | C23×C67 | C22×C4 |
| hypercenter | C2×C4×C8 | D67 |
| hypocenter | C67 | #1726 |
| LCS | G ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C22×C8 ◁ C2×C4×C8 | |
328 conjugacy classes of subgroups; only the 205 normal subgroups are shown.
This group has 328 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=d,b^2=e,c^2=f,d^2=g,c^b=cg,h^c=h^662192 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30742690, #30742692, #30746270, … |
| maximal normal subgroup of | #30742690, #30742692, #30746270, … |
| maximal quotient of | C8×#14523953, C8×#14523954, #30745186, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 4196);