Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | Aut(G) = C22×#23900 (order 1024) · Inn(G) = #120 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | #392 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
158 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 158 conjugacy classes of subgroups — too many to draw. Only its 34 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:
(2,6)(3,9)(7,13)(8,10)(11,15)(14,16);(1,2,4,8)(3,10,9,6)(5,7,12,14)(11,16,15,13);(3,11)(6,13)(9,15)(10,16)Polycyclic presentation:
2^7|b^2=f,d^2=f,b^a=bd,c^a=ce,d^a=df,d^b=df,e^b=eg,d^c=dg23 conjugacy classes — view the full character table.
| maximal subgroup of | #12202, #12204, #12205, … |
| maximal normal subgroup of | #12202, #12204, #12205, … |
| maximal quotient of | #7735, #7738, #7743, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 328);