Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | Aut(G) = #72216771 (order 1024) · Inn(G) = #357 · Out(G) = C22×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 | C2×D16 | 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 | |
39 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,10,7,12,6,23)(3,13,16,30,19,8,18,24)(5,20,15,27,22,28,21,11)(9,26,25,14,29,32,31,17);(2,8)(3,14)(4,6)(5,21)(9,27)(10,30)(11,29)(12,13)(15,22)(16,26)(17,19)(18,32)(20,25)(23,24)(28,31)Polycyclic presentation:
2^7|a^2=d,c^2=efg,d^2=g,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×#1419, #13625, #13658, … |
| maximal normal subgroup of | C2×#1419, #13625, #13658, … |
| maximal quotient of | #7385, #7420, #7444, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 151);