Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19713 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#19726 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3184 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2092 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)(5,12,16,18)(7,11,8,17)(10,14,13,15)(19,20,25,30)(21,27,32,33)(22,26,31,34)(23,28,24,29);(3,5,7,14,9,16,8,15)(4,10,17,18,6,13,11,12)(19,21,24,31,25,32,23,22)(20,26,28,33,30,34,29,27);(3,6,9,4)(5,13,16,10)(7,11,8,17)(12,15,18,14)(19,22,23,32,25,31,24,21)(20,27,29,34,30,33,28,26);(3,7,9,8)(4,11,6,17)(5,14,16,15)(10,12,13,18)(19,23,25,24)(20,28,30,29)(21,31,32,22)(26,27,34,33)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264558);