Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20158 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#20246 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3325 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1785 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,8)(6,14)(9,15)(10,17)(11,18)(12,16)(19,20,24,27)(21,28,29,34)(22,26,23,31)(25,32,33,30);(3,4)(5,11)(6,15)(7,9)(8,14)(10,12)(13,17)(16,18)(21,29)(25,33)(28,34)(30,32);(4,9)(8,15)(10,18)(11,17)(19,21)(20,25)(22,30)(23,32)(24,29)(26,34)(27,33)(28,31);(3,5,7,13)(4,10,9,18)(6,16,14,12)(8,17,15,11)(19,22,24,23)(20,26,27,31)(21,30,29,32)(25,34,33,28)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6266797);