Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C2×C4×C8×C67 ⋊ C2 |
| Sylow | C67 ⋊ #2922 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×D67 |
| commutator, socle | C22×C67 | C23×C4 |
| Frattini | C2×C4 | C23×D67 |
| Fitting | C2×C4×C8×C67 | C2 |
| hypercenter | C2×C4×C8 | D67 |
| hypocenter | C67 | #2922 |
| LCS | G ▷ C22×C67 ▷ C67 | |
| UCS | C1 ◁ C42 ◁ C2×C4×C8 | |
400 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 181 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=e,f^2=g,b^a=be,d^b=dg,h^b=h^662168 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30746521, #30746523, #30746526, … |
| maximal normal subgroup of | #30746521, #30746523, #30746526, … |
| maximal quotient of | #30853315, C4×#16863362, C4×#16863422, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 13152);