Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | C23 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
179 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,17)(11,19,21,15)(12,23,24,32)(14,27,26,18)(16,28,31,25)(20,35,36,40)(22,30,38,34)(29,37,33,39);(2,5)(6,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,29)(17,19)(18,30)(23,40)(24,36)(25,37)(26,38)(27,34)(28,39)(31,33)(32,35);(9,12,13,24)(10,16,17,31)(11,20,21,36)(14,28,26,25)(15,29,19,33)(18,32,27,23)(22,39,38,37)(30,35,34,40);(1,3,4,7)(2,6,5,8)(11,21)(15,19)(20,36)(22,38)(29,33)(30,34)(35,40)(37,39)Polycyclic presentation:
2^7|a^2=g,c^2=g,d^2=eg,f^2=g,b^a=be,c^a=cf,f^a=fg,d^b=dg,f^c=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#3127, #60557, #60566, … |
| maximal normal subgroup of | C2×#3127, #60557, #60566, … |
| maximal quotient of | #20745, #21127, #21135, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1859);