Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | Aut(G) = #72224662 (order 2048) · Inn(G) = D32 · Out(G) = C2×C16 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D64 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D32 |
| commutator, Frattini, cosocle | C32 | C22 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ G | |
| LECS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64);(2,64)(3,63)(4,62)(5,61)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,53)(14,52)(15,51)(16,50)(17,49)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(25,41)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)Polycyclic presentation:
2^7|c^2=de,d^2=ef,e^2=fg,f^2=g,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,c^b=cd,d^b=de,e^b=ef,f^b=fg35 conjugacy classes — view the full character table.
| maximal subgroup of | D128, SD256, C2×D64, … |
| maximal normal subgroup of | D128, SD256, C2×D64, … |
| maximal quotient of | #7537, #7545, D128, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 161);