Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) = #72252166 (order 4096) · Inn(G) = #121 · Out(G) = #2403 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
191 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6)(7,15)(8,13)(9,12)(10,14)(11,16)(17,20)(18,19);(2,4)(5,7)(6,11)(8,17)(9,16)(10,18)(12,19)(13,15)(14,20);(6,12)(7,16)(8,9)(13,14)(15,19)(17,18)Polycyclic presentation:
2^7|d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,c^b=cf,d^b=dg,f^b=fg,d^c=dg,e^c=eg,f^c=fg26 conjugacy classes — view the full character table.
| maximal subgroup of | #7398, #7400, #12108, … |
| maximal normal subgroup of | #7398, #7400, #12108, … |
| maximal quotient of | #9974, #9977, #9986, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 750);