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×#19989 |
| 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 | |
1764 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);(3,4,9,6)(5,13,16,18)(7,12,8,11)(10,17,14,15)(19,20)(21,24)(22,25)(23,27)(26,30)(28,31)(29,33)(32,34);(3,5)(4,10)(6,14)(7,15)(8,17)(9,16)(11,13)(12,18)(19,21,23,29)(20,24,27,33)(22,28,30,34)(25,31,26,32);(3,6,9,4)(5,14,16,10)(7,12,8,11)(13,15,18,17)(20,25)(22,30)(24,31)(26,27)(28,34)(32,33);(3,7)(4,11)(6,12)(8,9)(13,18)(15,17)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265576);