Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) = #72236620 (order 2048) · Inn(G) = #121 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×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 | |
148 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 148 conjugacy classes of subgroups — too many to draw. Only its 56 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:
(2,4)(5,6,10,14,11,13,9,7)(8,16,18,20,19,15,17,12);(1,2,3,4)(5,7,10,13,11,14,9,6)(8,15,18,20,19,16,17,12);(5,8,9,17,11,19,10,18)(6,12,14,16,13,20,7,15)Polycyclic presentation:
2^7|a^2=f,b^2=df,c^2=e,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dg,f^b=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1977, #21248, #21249, … |
| maximal normal subgroup of | C2×#1977, #21248, #21249, … |
| maximal quotient of | #9667, #9672, #9723, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 709);