Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×C22⋊C4 |
| commutator | C23 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | C22×#41 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
292 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)(5,6,9,14,10,13,8,12)(7,15,16,20,17,11,18,19);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,15)(14,20);(2,4)(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)Polycyclic presentation:
2^7|a^2=f,c^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,f^b=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #7306, #7309, #7312, … |
| maximal normal subgroup of | #7306, #7309, #7312, … |
| maximal quotient of | #8300, #8301, #8304, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 517);