Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19186 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#19214 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1902 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);(4,8)(6,9)(10,14)(11,15)(13,16)(17,18)(19,20)(21,27)(22,24)(23,31)(25,29)(26,30)(28,34)(32,33);(3,4,9,15,7,11,6,8)(5,10,16,18,12,17,13,14)(20,24)(25,33)(26,31)(27,34);(3,5)(4,10)(6,13)(7,12)(8,14)(9,16)(11,17)(15,18)(19,21,22,28)(20,25,24,33)(23,32,30,29)(26,34,31,27);(5,12)(10,17)(13,16)(14,18)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260896);