Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C24 |
| commutator, Frattini, socle, cosocle | C22 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 16 points:
(2,5)(6,8)(9,10)(11,13)(12,14)(15,16);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,13,14,16);(10,14)(13,16);(2,5)(6,8);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^7|b^2=g,e^2=f,b^a=bf,c^a=cg,d^b=df50 conjugacy classes — view the full character table.
| maximal subgroup of | #8438, #17508, #17512, … |
| maximal normal subgroup of | #8438, #17508, #17512, … |
| maximal quotient of | #34109, #34121, D4×#117, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2207);