Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) = #72252600 (order 4096) · Inn(G) = #120 · Out(G) = C2×D42 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| 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 | |
180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6,9,14,10,12,8,13)(7,15,18,11,17,20,16,19);(1,2,3,4)(5,7)(6,11)(8,18)(9,16)(10,17)(12,19)(13,20)(14,15);(6,12)(8,9)(11,19)(16,18)Polycyclic presentation:
2^7|a^2=f,b^2=df,d^2=g,f^2=g,b^a=bd,c^a=ce,e^a=eg,d^b=dg,e^b=eg,f^b=fg,d^c=dg,f^c=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #7487, #7495, #12761, … |
| maximal normal subgroup of | #7487, #7495, #12761, … |
| maximal quotient of | #8957, #8961, #8999, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 617);