Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D82 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#19994 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1848 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);(4,9)(5,7)(6,13)(10,14)(11,15)(17,18)(19,20)(21,24)(22,25)(23,27)(26,30)(28,31)(29,33)(32,34);(3,4)(5,9)(6,14)(7,15)(8,11)(10,12)(13,17)(16,18)(19,21,23,29)(20,24,27,33)(22,28,30,34)(25,31,26,32);(3,5,8,7)(4,9,11,15)(6,12,13,16)(10,17,18,14)(20,25)(22,30)(24,31)(26,27)(28,34)(32,33);(3,6,8,13)(4,10,11,18)(5,12,7,16)(9,17,15,14)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265581);