Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) = C24×#442 (order 1024) · Inn(G) = C2×D8 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23 |
| prime radical | C2×#135 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,11,15)(7,16,19,34)(8,13,10,25)(9,21,24,35)(12,27,30,18)(14,20,32,23)(17,33,26,31)(22,28,36,29);(5,7)(6,12)(8,17)(9,22)(10,26)(11,19)(13,28)(14,31)(15,30)(16,21)(18,23)(20,27)(24,36)(25,29)(32,33)(34,35);(2,4)(5,8,11,10)(6,13,15,25)(7,17,19,26)(9,20,24,23)(12,28,30,29)(14,21,32,35)(16,33,34,31)(18,22,27,36)Polycyclic presentation:
2^7|a^2=g,c^2=g,d^2=fg,f^2=g,b^a=bd,c^a=ce,d^a=df,f^a=fg,d^b=df,f^b=fg29 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2189, #33516, #33519, … |
| maximal normal subgroup of | C2×#2189, #33516, #33519, … |
| maximal quotient of | #12107, #12163, #12283, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 921);