Finite Group DatabasebetaFinite Group Databasebeta| order | 8128 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | M32×C127 ⋊ C2 |
| Sylow | C127 ⋊ #350 |
| subgroup | G/H | |
|---|---|---|
| center | C8 | C22⋊D127 |
| commutator | C4×C127 | C2×C8 |
| Frattini | C2×C8 | C2×D127 |
| Fitting | M32×C127 | C2 |
| socle | C2×C127 | C22⋊C8 |
| hypercenter | M32 | D127 |
| hypocenter | C127 | #350 |
| LCS | G ▷ C4×C127 ▷ C2×C127 ▷ C127 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ M32 | |
56 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,127|a^2=d,c^2=f,d^2=e,e^2=f,b^a=bc,g^a=g^126,c^b=cf,d^b=df,g^b=g^1261288 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #350×D127, C2×#22549661, #30640356, … |
| maximal normal subgroup of | #350×D127, C2×#22549661, #30640356, … |
| maximal quotient of | #30636655, #30636671, #30636684, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 334);