Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) = C26×D4 (order 512) · Inn(G) = C2×D4 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D4 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C22 | C4×D4 |
| prime radical | C2×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
112 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 112 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:
(2,3)(5,6,9,13,10,8,17,12)(7,11,15,19,16,14,20,18);(1,2,4,3)(5,7,9,15,10,16,17,20)(6,11,13,19,8,14,12,18);(1,3,4,2)(5,8)(6,10)(7,14)(9,12)(11,16)(13,17)(15,18)(19,20)Polycyclic presentation:
2^7|a^2=f,b^2=df,c^2=d,e^2=g,f^2=g,b^a=bd,c^a=ce,e^a=eg,e^c=eg38 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1929, #21070, #21071, … |
| maximal normal subgroup of | C2×#1929, #21070, #21071, … |
| maximal quotient of | #9541, #9542, #9543, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 661);