Finite Group DatabasebetaFinite Group Databasebeta| order | 8128 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×#41×C127 ⋊ C2 |
| Sylow | C127 ⋊ #532 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D127 |
| commutator, socle | C22×C127 | C24 |
| Frattini | C22 | C23×D127 |
| Fitting | C2×#41×C127 | C2 |
| hypercenter | C2×#41 | D127 |
| hypocenter | C127 | #532 |
| LCS | G ▷ C22×C127 ▷ C127 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#41 | |
310 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,127|c^2=f,d^2=e,b^a=be,c^a=cf,g^c=g^1261288 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30639110, #30639111, #30639113, … |
| maximal normal subgroup of | #30639110, #30639111, #30639113, … |
| maximal quotient of | #30641133, #30641189, #30641218, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 1095);