Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19050 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#19055 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3545 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2922 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);(4,9)(5,12)(6,7)(11,14)(13,15)(16,18)(19,20,23,26)(21,27,28,34)(22,30,31,25)(24,32,33,29);(3,4)(5,10)(6,14)(7,9)(8,11)(12,18)(13,17)(15,16)(22,31)(25,30)(27,34)(29,32);(3,5)(4,10)(6,12)(7,15)(8,13)(9,16)(11,17)(14,18)(19,21)(20,24)(22,32)(23,28)(25,34)(26,33)(27,30)(29,31);(5,13)(10,17)(12,15)(16,18)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258655);