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×#20013 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#127 | C23 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1610 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);(4,8)(5,11)(6,14)(9,12)(10,15)(17,18)(19,20,24,27)(21,25,23,31)(22,29,30,33)(26,32,34,28);(3,4,7,10)(5,12,11,17)(6,15,14,8)(9,16,18,13)(19,21,24,23)(20,25,27,31)(22,28,30,32)(26,33,34,29);(19,22,24,30)(20,26,27,34)(21,28,23,32)(25,33,31,29);(3,5,7,11)(4,9,10,18)(6,16,14,13)(8,17,15,12)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265603);