Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1705 |
| automorphism | Aut(G) = C2×#72260766 (order 12288) · Inn(G) = #127 · Out(G) = C2×#392⋊C3 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #127 |
| commutator, Frattini, cosocle | C42 | C23 |
| prime radical | #127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
89 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9)(4,13)(5,7)(6,11)(8,10)(12,15)(14,16)(17,18)(19,25)(20,22)(21,29)(23,27)(24,28)(26,31)(30,32);(1,3,10,16,5,12,4,11)(2,6,13,15,7,14,8,9)(18,22)(23,30)(24,29)(25,31);(2,7)(6,14)(8,13)(9,15)(17,19,21,27,20,26,28,32)(18,23,24,25,22,30,29,31)Polycyclic presentation:
2^7|b^2=dfg,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.
| Fitting subgroup of | #1711⋊C3 |
| maximal subgroup of | C2×#1711, #19600, #19627, … |
| maximal normal subgroup of | C2×#1711, #19600, #19627, … |
| maximal quotient of | #8817, #9821, #9822, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 443);