Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) = #72236881 (order 1024) · Inn(G) = #121 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #121 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C2 | #389 |
| prime radical | C23 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
92 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,10,7,11,5,21)(3,12,14,8,16,23,15,24)(4,17,19,25,20,31,18,9)(13,22,27,30,29,32,28,26);(1,3,6,14,7,16,5,15)(2,8,10,23,11,24,21,12)(4,13,19,27,20,29,18,28)(9,22,17,30,25,32,31,26);(1,4,5,18,7,20,6,19)(2,9,10,17,11,25,21,31)(3,13,15,28,16,29,14,27)(8,22,23,30,24,32,12,26)Polycyclic presentation:
2^7|a^2=f,b^2=f,c^2=e,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dg32 conjugacy classes — view the full character table.
| maximal subgroup of | #12942, #12947, #12949, … |
| maximal normal subgroup of | #12942, #12947, #12949, … |
| maximal quotient of | #9607, #9608, #9609, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 685);