Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1705 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | #127 |
| commutator, Frattini, cosocle | C42 | C23 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,7)(3,9,11,16)(4,12,13,8)(6,14,15,10)(17,18)(19,25)(20,22)(21,29)(23,27)(24,28)(26,31)(30,32);(1,3,4,10,5,11,13,14)(2,6,8,9,7,15,12,16)(18,22)(23,30)(24,29)(25,31);(2,7)(6,15)(8,12)(9,16)(17,19,21,27,20,26,28,32)(18,23,24,25,22,30,29,31)Polycyclic presentation:
2^7|a^2=f,b^2=dg,c^2=ef,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg38 conjugacy classes — view the full character table.
| maximal subgroup of | #7093, #12449, #12450, … |
| maximal normal subgroup of | #7093, #12449, #12450, … |
| maximal quotient of | #8816, #9821, #9823, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 442);