Finite Group DatabasebetaFinite Group Databasebeta| order | 8128 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C127 ⋊ #552 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D127 |
| commutator, socle | C22×C127 | C24 |
| Fitting | #123×C127 | C2 |
| hypercenter | #123 | D127 |
| hypocenter | C127 | #552 |
| LCS | G ▷ C22×C127 ▷ C127 | |
| UCS | C1 ◁ C22 ◁ #123 | |
214 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 214 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,127|a^2=e,b^2=f,c^2=e,b^a=be,c^a=cf,d^b=de,g^b=g^126,g^c=g^126907 conjugacy classes — view the full character table.
| maximal subgroup of | #552×D127, C2×#22550563, #30652281, … |
| maximal normal subgroup of | #552×D127, C2×#22550563, #30652281, … |
| maximal quotient of | #30641588, #30641592, #30641594, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 1236);