Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | Aut(G) = C24×#442 (order 1024) · Inn(G) = C2×D8 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
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 | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(6,10)(7,14)(8,15)(11,16)(12,17)(13,20)(18,19);(5,6,11,20,8,13,7,10,9,12,14,19,15,18,16,17);(2,4)(6,12)(10,17)(13,18)(19,20)Polycyclic presentation:
2^7|b^2=dfg,d^2=fg,f^2=g,b^a=bd,c^a=ce,d^a=df,f^a=fg,c^b=cg26 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2220, #33604, #33605, … |
| maximal normal subgroup of | C2×#2220, #33604, #33605, … |
| maximal quotient of | #12435, #12444, #12456, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 952);