Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C210 (order 1024) · Inn(G) = C23 · Out(G) = C27 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C23 |
| commutator | C22 | C22×C8 |
| socle, prime radical | C23 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
120 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,15,17,20,16,19);(6,9)(10,12)(13,15,18,20,17,19,16,14);(2,4)(5,7,8,11)(6,10,9,12)(13,16,17,18)(14,19,20,15)Polycyclic presentation:
2^7|a^2=f,b^2=f,c^2=d,f^2=g,b^a=bd,c^a=ce56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1916, #21020, #21022, … |
| maximal normal subgroup of | C2×#1916, #21020, #21022, … |
| maximal quotient of | #9533, #9534, #9535, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 648);