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×#20005 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#550 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1670 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);(3,4,9,11)(5,12,15,16)(6,7,8,17)(10,18,14,13)(19,20,23,27)(21,24,29,33)(22,25,30,26)(28,31,34,32);(3,5,9,15)(4,10,11,14)(6,13,8,18)(7,16,17,12)(19,21,23,29)(20,24,27,33)(22,28,30,34)(25,31,26,32);(3,6,9,8)(4,7,11,17)(5,13,15,18)(10,16,14,12)(20,25)(22,30)(24,31)(26,27)(28,34)(32,33);(3,7,9,17)(4,6,11,8)(5,14,15,10)(12,18,16,13)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265593);