Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | Aut(G) = #72236189 (order 1024) · Inn(G) = #394 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #394 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | C2×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
137 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 137 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:
(2,7)(3,10)(4,5)(8,15)(9,13)(11,12);(1,2)(3,8)(4,13)(5,7)(6,9)(10,14)(11,16)(12,15);(1,3,4,11,6,12,5,10)(2,8,7,14,9,15,13,16)Polycyclic presentation:
2^7|c^2=d,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=cf,d^b=dg,f^c=fg23 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2048, #23525, #24295, … |
| maximal normal subgroup of | C2×#2048, #23525, #24295, … |
| maximal quotient of | #10440, #10441, #10442, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 780);