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) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23 |
| prime radical | #466 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
71 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(10,14)(11,17)(13,20)(15,23)(16,19)(18,24)(21,22);(2,5)(6,8)(9,10,12,14)(11,15,18,22)(13,16,19,20)(17,21,24,23);(1,3,4,7)(2,6,5,8)(9,11,12,18)(10,15,14,22)(13,21,19,23)(16,24,20,17)Polycyclic presentation:
2^7|b^2=g,c^2=d,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×#2234, #33645, #33647, … |
| maximal normal subgroup of | C2×#2234, #33645, #33647, … |
| maximal quotient of | #12868, #12921, #12992, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 966);