Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | Aut(G) = #72232235 (order 6144) · Inn(G) = #95 · Out(G) = C2×GL(2,ℤ/4) |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #95 |
| commutator, prime radical | C23 | C42 |
| Frattini, cosocle | C2×C42 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
73 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3,6,4,5,7,8)(10,13)(11,15);(2,5)(3,7)(9,10,11,14,12,13,15,16)Polycyclic presentation:
2^7|a^2=d,b^2=e,d^2=f,e^2=g,b^a=bc,c^a=cf,e^a=eg,c^b=cg,d^b=df26 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 41);