Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | Aut(G) = C22×#12067 (order 1024) · Inn(G) = D16 · Out(G) = C22×C8 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | D16 |
| commutator | C16 | C2×C4 |
| Frattini, cosocle | C2×C16 | C22 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
37 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2,3,4)(6,11)(7,15)(8,17)(9,21)(12,27)(13,30)(14,26)(16,20)(18,34)(19,24)(22,28)(23,25)(29,35)(31,32)(33,36);(5,6,10,14)(7,16,19,33)(8,20,22,36)(9,23,21,13)(11,24,26,15)(12,28,29,17)(18,35,31,27)(25,32,30,34)Polycyclic presentation:
2^7|a^2=d,b^2=g,c^2=ef,e^2=fg,f^2=g,b^a=bc,c^a=ce,e^a=ef,f^a=fg,c^b=ce,e^b=ef,f^b=fg38 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1416, #13615, #13616, … |
| maximal normal subgroup of | C2×#1416, #13615, #13616, … |
| maximal quotient of | #7384, #7419, #7445, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 148);