Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C67 ⋊ #1451 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C42 | C22×D67 |
| commutator, socle | C22×C67 | C2×C42 |
| Fitting | C2×C4×C8×C67 | C2 |
| hypercenter | C2×C4×C8 | D67 |
| hypocenter | C67 | #1451 |
| LCS | G ▷ C22×C67 ▷ C67 | |
| UCS | C1 ◁ C42 ◁ C2×C4×C8 | |
208 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 125 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,d^2=f,e^2=g,b^a=bg,c^a=cf,h^a=h^662168 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30742724, #30742725, #30742730, … |
| maximal normal subgroup of | #30742724, #30742725, #30742730, … |
| maximal quotient of | #30745193, #30745213, #30746257, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 2799);