Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) = C22×AGL(1,ℤ/16) (order 512) · Inn(G) = D8 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | D8 |
| commutator | C8 | C22×C4 |
| Frattini, cosocle | C2×C8 | C23 |
| socle | C2 | C4×D8 |
| prime radical | #508 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
73 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,10,7,12,22,11)(3,13,15,24,17,30,16,32)(4,9,18,27,19,28,5,20)(8,23,25,31,26,21,29,14);(1,3)(2,8)(4,14)(5,21)(6,15)(7,17)(9,24)(10,25)(11,29)(12,26)(13,20)(16,22)(18,23)(19,31)(27,30)(28,32);(1,4,6,18,7,19,22,5)(2,9,10,27,12,28,11,20)(3,14,15,23,17,31,16,21)(8,24,25,30,26,32,29,13)Polycyclic presentation:
2^7|a^2=e,c^2=e,d^2=fg,e^2=g,f^2=g,b^a=bd,d^a=df,f^a=fg,d^b=df,f^b=fg44 conjugacy classes — view the full character table.
| maximal subgroup of | #7387, #7388, #12046, … |
| maximal normal subgroup of | #7387, #7388, #12046, … |
| maximal quotient of | C8×D16, C8×SD32, C8×Q32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 910);