Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | Aut(G) = #72249436 (order 4096) · Inn(G) = C2×D8 · Out(G) = #2010 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,16,22)(11,18,26,37)(12,17,28,35)(13,20,30,32)(14,19,23,38)(15,31,29,21)(24,36,39,34)(25,40,27,33);(2,5)(6,8)(9,11,23,39,15,25,13,12,16,26,14,24,29,27,30,28)(10,17,32,40,21,34,19,18,22,35,20,33,31,36,38,37);(1,3,4,7)(2,6,5,8)(9,12,13,27,29,39,23,26,16,28,30,25,15,24,14,11)(10,18,19,36,31,40,32,35,22,37,38,34,21,33,20,17)Polycyclic presentation:
2^7|a^2=g,b^2=eg,c^2=d,d^2=fg,e^2=f,f^2=g,b^a=bd,c^a=ce,d^a=df,e^a=efg,f^a=fg,c^b=cg26 conjugacy classes — view the full character table.
| maximal subgroup of | #12350, #12351, #12362, … |
| maximal normal subgroup of | #12350, #12351, #12362, … |
| maximal quotient of | #12890, #13150, #13207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 977);