Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | Aut(G) = #72231721 (order 2048) · Inn(G) = #120 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator, Frattini, cosocle | C42 | C23 |
| prime radical | #555 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
109 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,14,17)(11,18,20,23)(12,16,13,21)(15,22,24,19);(3,7)(5,8)(9,11,14,20)(10,15,17,24)(12,19,13,22)(16,23,21,18);(1,2,4,6)(3,8,7,5)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)Polycyclic presentation:
2^7|a^2=f,b^2=f,c^2=fg,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,d^b=df,e^b=eg,d^c=dg,e^c=eg26 conjugacy classes — view the full character table.
| maximal subgroup of | #7521, #13682, C2×#1634, … |
| maximal normal subgroup of | #7521, #13682, C2×#1634, … |
| maximal quotient of | #7758, #7760, #7785, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 366);