Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | Aut(G) = C2×AGL(1,ℤ/32) (order 1024) · Inn(G) = D16 · Out(G) = C22×C8 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D16 |
| commutator | C16 | C2×C4 |
| Frattini, cosocle | C2×C16 | C22 |
| socle | C2 | #366 |
| prime radical | C22 | Q32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
25 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,8,13,7,29)(3,14,17,39,20,41,19,51)(4,21,23,52,25,54,24,55)(6,27,26,37,28,56,22,12)(9,30,31,47,33,46,57,16)(10,34,35,44,36,43,53,15)(18,49,48,61,50,64,45,40)(32,42,38,58,59,63,60,62);(1,3,15,42,6,18,23,30,5,17,34,58,26,48,25,47,8,20,44,63,28,50,24,46,7,19,43,62,22,45,4,16)(2,9,21,40,12,32,53,51,29,57,55,64,56,60,36,41,13,33,54,61,37,59,35,39,11,31,52,49,27,38,10,14)Polycyclic presentation:
2^7|a^2=d,b^2=ce,c^2=ef,d^2=g,e^2=fg,f^2=g,b^a=bc,c^a=ce,e^a=ef,f^a=fg38 conjugacy classes — view the full character table.
| maximal subgroup of | #7539, #7541, #7542, … |
| maximal normal subgroup of | #7539, #7541, #7542, … |
| maximal quotient of | #7460, #7461, #7472, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 157);