Finite Group DatabasebetaFinite Group Databasebeta| order | 8128 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C127 · Schur multiplier (H2) = C24 |
| direct product | Q82×C127 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C127 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C127 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ G | |
212 conjugacy classes of subgroups.
3175 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C127×#1623, C127×#1633, C127×#1645, … |
| maximal normal subgroup of | C127×#1623, C127×#1633, C127×#1645, … |
| maximal quotient of | Q8×C4⋊C4×C127, C127×#2369, C127×#2451, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 239);