Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | Aut(G) = #72215179 (order 1024) · Inn(G) = #120 · Out(G) = #120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | #120 |
| commutator, Frattini, cosocle | C42 | C23 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
86 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,11)(3,12,14,15)(4,13,8,16)(5,10,6,9)(17,18,22,20)(19,25,28,32)(21,29,31,24)(23,30,26,27);(1,3)(2,8)(4,11)(5,16)(6,13)(7,14)(9,15)(10,12)(17,19,22,28)(18,23,20,26)(21,30,31,27)(24,32,29,25);(1,4,7,8)(2,3,11,14)(5,15,6,12)(9,13,10,16)(17,20,22,18)(19,26,28,23)(21,24,31,29)(25,30,32,27)Polycyclic presentation:
2^7|a^2=f,b^2=g,c^2=f,d^2=f,e^2=fg,b^a=bd,c^a=ce,d^a=df,e^a=efg,d^b=df,e^b=eg,d^c=dg,e^c=efg20 conjugacy classes — view the full character table.
| maximal subgroup of | #7526, C2×#1663, #18990, … |
| maximal normal subgroup of | #7526, C2×#1663, #18990, … |
| maximal quotient of | #7763, #7768, #7772, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 395);