Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | Aut(G) = C2×#72233929 (order 4096) · Inn(G) = #127 · Out(G) = C2×#392 |
| 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 | |
82 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8)(3,9,10,16)(4,12,13,7)(6,14,15,11)(17,18)(19,25)(20,22)(21,29)(23,27)(24,28)(26,31)(30,32);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16)(18,22)(23,30)(24,29)(25,31);(3,10)(6,15)(9,16)(11,14)(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=e,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,c^b=cf26 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1718, #19613, #19620, … |
| maximal normal subgroup of | C2×#1718, #19613, #19620, … |
| maximal quotient of | #8821, #9826, #9828, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 450);