Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#543 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C24 |
| commutator, Frattini, cosocle | C22 | C25 |
| prime radical | C25 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4)(5,9)(6,7)(8,10)(11,12,15,16)(13,18,17,14);(4,7)(8,9)(12,16)(13,17);(3,4)(5,8)(6,7)(9,10)(11,13,15,17)(12,14,16,18);(3,5)(4,8)(6,10)(7,9)(11,14)(12,17)(13,16)(15,18)44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2188);