Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) = C24×D42 (order 1024) · Inn(G) = #121 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | C23 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,9,16,11,15,10,14)(7,13,17,12,8,19,18,20);(1,2)(3,4)(5,7,9,17,11,8,10,18)(6,12,14,13,15,20,16,19);(2,4)(5,8,11,7)(6,13,15,19)(9,17,10,18)(12,16,20,14)Polycyclic presentation:
2^7|a^2=de,b^2=d,c^2=g,d^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,c^b=cf,d^c=dg,f^c=fg26 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2064, #23601, #23603, … |
| maximal normal subgroup of | C2×#2064, #23601, #23603, … |
| maximal quotient of | #10588, #10590, #10592, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 796);