Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1705 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #127 |
| commutator, Frattini, cosocle | C42 | C23 |
| prime radical | #493 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,3)(4,5)(7,8)(9,10)(11,17)(12,20)(13,15)(14,18)(16,21)(19,23)(22,24);(1,2,5,7,6,8,4,3)(9,11,12,18,13,19,21,24)(10,14,16,17,15,22,20,23);(1,3,4,8,6,7,5,2)(10,15)(14,22)(16,20)(17,23)Polycyclic presentation:
2^7|b^2=dg,c^2=ef,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg38 conjugacy classes — view the full character table.
| Frattini subgroup of | #94820, #94821, #94827, … |
| maximal subgroup of | #7092, #7530, #7531, … |
| maximal normal subgroup of | #7092, #7530, #7531, … |
| maximal quotient of | #8816, #9822, #9824, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 441);