Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | Aut(G) order 983040 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#564 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C24 |
| commutator, Frattini, cosocle | C22 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
448 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(5,11)(6,14)(8,12)(9,16)(19,20,23,26)(21,24,29,27)(22,30,31,25)(28,33,34,32);(3,4,7,10)(5,12,11,8)(6,9,14,16)(13,18,17,15)(19,20)(21,27)(22,30)(23,26)(24,29)(25,31)(28,32)(33,34);(3,5,7,11)(4,8,10,12)(6,13,14,17)(9,15,16,18)(19,21)(20,24)(22,28)(23,29)(25,32)(26,27)(30,33)(31,34);(3,6)(4,9)(5,13)(7,14)(8,15)(10,16)(11,17)(12,18)(19,22,23,31)(20,25,26,30)(21,28,29,34)(24,32,27,33)38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2262);