Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1408 |
| automorphism | Aut(G) = #72216601 (order 1024) · Inn(G) = #353 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #353 |
| commutator | C42 | C2×C4 |
| Frattini, cosocle | #127 | C22 |
| prime radical | #493 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2)(3,10,11,6)(4,8,14,7)(5,9)(12,15,13,16);(1,3)(2,6)(4,13)(5,12)(7,15)(8,10)(9,16)(11,14)Polycyclic presentation:
2^7|a^2=d,c^2=fg,e^2=g,b^a=bc,c^a=ce,e^a=ef,f^a=fg,c^b=cfg,d^b=def,e^b=eg,d^c=dfg,e^d=eg17 conjugacy classes — view the full character table.
| maximal subgroup of | #7515, #7516, #7529, … |
| maximal normal subgroup of | #7515, #7516, #7529, … |
| maximal quotient of | #7354, #7370, #7402, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 140);