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) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D16 |
| commutator, Frattini, cosocle | C16 | C23 |
| socle | C2 | C2×D16 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
53 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,14)(4,9)(5,23)(6,27)(7,13)(8,16)(10,25)(11,21)(12,22)(15,41)(17,47)(18,49)(19,42)(20,50)(24,38)(26,56)(28,35)(29,34)(30,43)(31,57)(32,46)(33,52)(36,62)(37,59)(39,54)(40,45)(44,61)(48,63)(51,60)(53,55)(58,64);(1,3)(2,8)(4,15)(5,24)(6,28)(7,19)(9,30)(10,35)(11,38)(12,40)(13,32)(14,23)(16,25)(17,21)(18,22)(20,51)(26,58)(27,31)(29,44)(33,47)(34,60)(36,63)(37,61)(39,64)(41,50)(42,56)(43,52)(45,62)(46,59)(48,54)(49,55)(53,57);(1,4,7,22)(2,9,13,12)(3,15,19,18)(5,20,26,53)(6,21,29,54)(8,30,32,40)(10,33,37,36)(11,34,39,27)(14,41,42,49)(16,43,46,45)(17,44,48,28)(23,50,56,55)(24,51,58,57)(25,52,59,62)(31,38,60,64)(35,47,61,63)Polycyclic presentation:
2^7|c^2=g,d^2=ef,e^2=fg,f^2=g,b^a=bd,d^a=de,e^a=ef,f^a=fg,d^b=de,e^b=ef,f^b=fg38 conjugacy classes — view the full character table.
| maximal subgroup of | #7539, #7540, #13655, … |
| maximal normal subgroup of | #7539, #7540, #13655, … |
| maximal quotient of | #13615, #13633, C4×D32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 994);