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 ⋊ C4×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D127 |
| commutator | C2×C127 | C23×C4 |
| Frattini | C22 | C23×D127 |
| Fitting | #117×C127 | C2 |
| socle | C22×C127 | C24 |
| hypercenter | #117 | D127 |
| hypocenter | C127 | C4×#41 |
| LCS | G ▷ C2×C127 ▷ C127 | |
| UCS | C1 ◁ C2×C4 ◁ #117 | |
310 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 155 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,c^2=f,d^2=e,b^a=be,g^c=g^1261300 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30638534, #30638539, #30638749, … |
| maximal normal subgroup of | #30638534, #30638539, #30638749, … |
| maximal quotient of | #30641069, #117×Dic127, C4×(C22⋊D127)⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 1017);