Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) = C24×D42 (order 1024) · Inn(G) = #121 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #121 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | C23 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,4,3)(6,10)(7,13)(8,15)(11,18)(12,17)(16,19);(1,3,4,2)(5,6,8,10,9,12,15,17)(7,11,14,16,13,19,20,18);(5,7,8,14,9,13,15,20)(6,11,10,16,12,19,17,18)Polycyclic presentation:
2^7|a^2=f,b^2=df,c^2=d,d^2=g,b^a=bd,c^a=ce,d^a=dg,e^c=eg32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1930, #21075, #21076, … |
| maximal normal subgroup of | C2×#1930, #21075, #21076, … |
| maximal quotient of | #9549, #9551, #9561, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 662);