Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | Aut(G) = S4×AGL(1,ℤ/16) (order 3072) · Inn(G) = C22×D8 · Out(G) = C2×C4×D3 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×D8 |
| commutator, Frattini, cosocle | C8 | C24 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #131 ◁ G | |
| LECS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
185 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,8)(4,7)(5,18)(6,11)(9,15)(10,16)(12,22)(13,19)(14,28)(17,21)(20,29)(23,27)(24,26)(25,30)(31,32);(2,7)(4,15)(5,16)(8,19)(9,10)(11,21)(13,27)(14,26)(17,29)(18,20)(22,30)(23,24)(25,32)(28,31);(1,3)(2,8)(4,13)(5,14)(6,12)(7,19)(9,23)(10,24)(11,22)(15,27)(16,26)(17,25)(18,28)(20,31)(21,30)(29,32);(3,12)(8,22)(13,25)(14,26)(19,30)(23,31)(24,28)(27,32)Polycyclic presentation:
2^7|e^2=fg,f^2=g,b^a=be,e^a=ef,f^a=fg,e^b=ef,f^b=fg,d^c=dg32 conjugacy classes — view the full character table.
| Fitting subgroup of | #3415⋊C3, #11711666, #11711668, … |
| maximal subgroup of | #13669, #13671, #33854, … |
| maximal normal subgroup of | #13669, #13671, #33854, … |
| maximal quotient of | #33381, #33410, #33454, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2147);