Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) = #72238359 (order 2048) · Inn(G) = #121 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #121 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | C2×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,10,13,11,8,9,19)(7,14,17,15,18,20,16,12);(5,7,9,16,11,18,10,17)(6,12,13,14,8,15,19,20);(2,4)(5,8)(6,11)(7,15)(9,19)(10,13)(12,18)(14,17)(16,20)Polycyclic presentation:
2^7|a^2=f,b^2=d,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^c=eg,f^c=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #12752, #12833, #12933, … |
| maximal normal subgroup of | #12752, #12833, #12933, … |
| maximal quotient of | #8604, #8607, #8626, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 547);