Finite Group DatabasebetaFinite Group Databasebeta| order | 8128 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | #117×C127 ⋊ C2 |
| Sylow | C127 ⋊ #520 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D127 |
| commutator | C2×C127 | C23×C4 |
| Fitting | #117×C127 | C2 |
| socle | C22×C127 | C24 |
| hypercenter | #117 | D127 |
| hypocenter | C127 | #520 |
| LCS | G ▷ C2×C127 ▷ C127 | |
| UCS | C1 ◁ C22 ◁ #117 | |
294 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 151 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=f,d^2=e,b^a=be,d^a=de,c^b=ce,g^c=g^1261294 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30638953, #30638957, #30638960, … |
| maximal normal subgroup of | #30638953, #30638957, #30638960, … |
| maximal quotient of | #30641084, #30641086, #30641089, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 1029);