Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) = C24×D42 (order 1024) · Inn(G) = #121 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #121 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
116 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 48 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) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,4,3)(6,10)(7,14)(8,11)(12,19)(13,17)(16,20);(1,3,4,2)(5,6,11,17,9,13,8,10)(7,12,18,19,14,20,15,16);(5,7,8,15,9,14,11,18)(6,12,10,16,13,20,17,19)Polycyclic presentation:
2^7|a^2=f,b^2=dfg,c^2=d,d^2=g,b^a=bd,c^a=ce,d^a=dg,e^c=eg32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1931, #21075, #21076, … |
| maximal normal subgroup of | C2×#1931, #21075, #21076, … |
| maximal quotient of | #9550, #9552, #9563, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 663);