Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C67 ⋊ #2398 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D67 |
| commutator, socle | C23×C67 | C24 |
| Frattini | C23 | C23×D67 |
| Fitting | C2×C4×D4×C67 | C2 |
| hypercenter | C2×C4×D4 | D67 |
| hypocenter | C67 | #2398 |
| LCS | G ▷ C23×C67 ▷ C67 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×D4 | |
568 conjugacy classes of subgroups; only the 161 normal subgroups are shown.
This group has 568 conjugacy classes of subgroups — too many to draw. Only its 161 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=e,b^2=e,d^2=ef,b^a=be,c^a=cf,c^b=cg,h^b=h^661364 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30750898, #30750900, #30750912, … |
| maximal normal subgroup of | #30750898, #30750900, #30750912, … |
| maximal quotient of | #30777534, #30777569, #30777588, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 7794);