Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C2×#394108 (order 1024) · Inn(G) = D4 · Out(G) = C2×#392 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | D4 |
| commutator | C4 | C4×C8 |
| Frattini, cosocle | C22×C8 | C22 |
| socle | C22 | C4⋊C8 |
| prime radical | C23 | M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
74 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2)(3,4)(5,6,8,13,9,12,16,19,10,14,17,20,7,11,15,18);(2,4)(6,11,14,12)(13,18,20,19)Polycyclic presentation:
2^7|a^2=d,b^2=e,c^2=g,d^2=f,f^2=g,b^a=bc,c^a=cg,e^a=eg56 conjugacy classes — view the full character table.
| maximal subgroup of | #7107, #7108, #7114, … |
| maximal normal subgroup of | #7107, #7108, #7114, … |
| maximal quotient of | #7138, #7143, #7146, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 108);