Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | Aut(G) = C2×#63011 (order 512) · Inn(G) = #124 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| socle | C2 | #393 |
| prime radical | C2×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
122 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 122 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,10,12,16)(4,8)(5,14)(6,9)(7,13,15,11);(3,11,12,13)(7,10,15,16);(1,3)(2,7)(4,11)(5,13)(6,12)(8,16)(9,15)(10,14)Polycyclic presentation:
2^7|a^2=d,b^2=d,e^2=g,f^2=g,b^a=bd,c^a=ce,f^a=fg,c^b=cf,d^c=dg,e^c=eg,f^c=fg26 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2031, #23472, #23833, … |
| maximal normal subgroup of | C2×#2031, #23472, #23833, … |
| maximal quotient of | #10362, #10363, #10364, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 763);