Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) = #72237167 (order 2048) · Inn(G) = C2×D8 · Out(G) = #525 |
| 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 | #466 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
113 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(6,10)(7,13)(8,15)(11,20)(12,19)(14,17)(16,18);(5,6,9,12)(7,14,16,20)(8,17,15,11)(10,18,19,13);(2,4)Polycyclic presentation:
2^7|b^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 | #12105, #12107, #12111, … |
| maximal normal subgroup of | #12105, #12107, #12111, … |
| maximal quotient of | #12103, #12159, #12279, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 917);