Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | Aut(G) = C24×D42 (order 1024) · Inn(G) = #121 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #121 |
| commutator, Frattini, cosocle | C42 | C23 |
| prime radical | C2×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
83 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,5,7)(3,8,6,4)(9,10)(11,13)(12,14)(15,16);(1,3,4,2,5,6,8,7)(10,13)(14,15);(10,14)(12,16)(13,15)Polycyclic presentation:
2^7|a^2=f,b^2=dg,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,e^c=eg29 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1679, #19110, #19124, … |
| maximal normal subgroup of | C2×#1679, #19110, #19124, … |
| maximal quotient of | #8646, #9217, #9219, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 411);