Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | Aut(G) = #72224639 (order 1024) · Inn(G) = D32 · Out(G) = C16 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| semi dihedral | SD128 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D32 |
| commutator, Frattini, cosocle | C32 | C22 |
| prime radical | D32 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ G | |
| LECS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
19 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);(2,32)(3,63)(4,30)(5,61)(6,28)(7,59)(8,26)(9,57)(10,24)(11,55)(12,22)(13,53)(14,20)(15,51)(16,18)(17,49)(19,47)(21,45)(23,43)(25,41)(27,39)(29,37)(31,35)(34,64)(36,62)(38,60)(40,58)(42,56)(44,54)(46,52)(48,50)Polycyclic presentation:
2^7|a^2=g,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 | C2×SD128, #13741, #13742, … |
| maximal normal subgroup of | C2×SD128, #13741, #13742, … |
| maximal quotient of | #7537, #7538, #7546, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 162);