Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | Aut(G) = #483875 (order 512) · Inn(G) = #357 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #357 |
| commutator | C16 | C2×C4 |
| Frattini, cosocle | C2×C16 | C22 |
| prime radical | #508 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
37 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(2,7,8,23)(3,11)(4,16)(5,18)(9,27,26,31)(10,28,20,25)(12,17,24,22)(13,19)(14,32)(15,21)(29,30);(1,2)(3,12)(4,17)(5,20)(6,8)(7,21)(9,16)(10,18)(11,23)(13,26)(14,27)(15,24)(19,22)(25,30)(28,32)(29,31)Polycyclic presentation:
2^7|a^2=d,c^2=efg,e^2=fg,f^2=g,b^a=bc,c^a=ce,e^a=ef,f^a=fg,c^b=ceg,d^b=dg,e^b=ef,f^b=fg26 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1418, #13622, #13657, … |
| maximal normal subgroup of | C2×#1418, #13622, #13657, … |
| maximal quotient of | #7385, #7386, #7418, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 150);