Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) = C22×#511771 (order 2048) · Inn(G) = #120 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | #537 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
185 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,10,14)(7,15,17,20)(8,13,9,12)(11,18,19,16);(1,2)(3,4)(5,7)(6,11)(8,18)(9,16)(10,17)(12,20)(13,15)(14,19);(2,4)(5,7)(6,11)(8,16)(9,18)(10,17)(12,15)(13,20)(14,19)Polycyclic presentation:
2^7|a^2=g,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=cf,d^b=dg,e^c=eg26 conjugacy classes — view the full character table.
| maximal subgroup of | #7409, #12098, #12100, … |
| maximal normal subgroup of | #7409, #12098, #12100, … |
| maximal quotient of | #9934, #9935, #9936, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 739);