Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = #72260440 (order 6144) · Inn(G) = C22 · Out(G) = GL(2,ℤ/8) |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C42 | C22 |
| commutator | C2 | C82 |
| socle, prime radical | C23 | C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2)(3,4)(5,6,7,9,8,10,11,12);(2,4)(13,14,15,17,16,18,19,20)Polycyclic presentation:
2^7|a^2=d,b^2=e,d^2=f,e^2=g,b^a=bc80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 5);