Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19186 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#19215 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C24 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,11)(6,8)(7,15)(9,13)(10,14)(12,18)(16,17)(19,20,24,27)(21,28,26,23)(22,29,30,34)(25,31,33,32);(4,8)(9,17)(10,15)(11,18)(19,21,23,20,24,26,28,27)(22,29,31,25,30,34,32,33);(3,5,6,12)(4,9,8,17)(7,16,14,13)(10,18,15,11)(19,22,24,30)(20,25,27,33)(21,29,26,34)(23,31,28,32);(22,30)(25,33)(29,34)(31,32)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260897);