Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) = C24×#442 (order 1024) · Inn(G) = C2×D8 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,12,15)(11,19,20,37)(13,25,23,38)(14,28,24,18)(16,32,30,21)(17,34,31,27)(22,39,29,35)(26,36,40,33);(1,3,4,8)(2,5,6,7)(10,15)(16,30)(17,31)(18,28)(19,37)(25,38)(33,36)(35,39);(9,11,12,20)(10,16,15,30)(13,26,23,40)(14,29,24,22)(17,35,31,39)(18,36,28,33)(19,38,37,25)(21,34,32,27)Polycyclic presentation:
2^7|a^2=d,b^2=dg,c^2=g,e^2=fg,f^2=g,b^a=bd,c^a=ce,e^a=ef,f^a=fg,e^c=ef,f^c=fg29 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2229, #33626, #33628, … |
| maximal normal subgroup of | C2×#2229, #33626, #33628, … |
| maximal quotient of | #12814, #12936, #12938, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 961);