Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | Aut(G) = C2×#72233929 (order 4096) · Inn(G) = #127 · Out(G) = C2×#392 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #127 |
| commutator, Frattini, cosocle | C42 | C23 |
| prime radical | #493 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,5)(3,7)(6,8)(10,15)(11,16)(12,13);(1,2,3,5,4,6,7,8)(9,10,13,11,14,16,12,15);(2,6)(5,8)(9,11,13,16,14,15,12,10)Polycyclic presentation:
2^7|b^2=dg,c^2=e,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,c^b=cf26 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1717, #19613, #19619, … |
| maximal normal subgroup of | C2×#1717, #19613, #19619, … |
| maximal quotient of | #8821, #9827, #9829, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 449);