Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19713 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C8 |
| direct product | C2×#19729 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2296 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)(6,9)(10,14)(11,15)(13,16)(17,18)(19,20)(21,29)(22,25)(23,33)(24,28)(26,31)(27,30)(32,34);(3,4,9,15,7,11,6,8)(5,10,16,18,12,17,13,14)(19,21,30,32,24,31,23,22)(20,25,33,26,28,34,27,29);(3,5,7,12)(4,10,11,17)(6,13,9,16)(8,14,15,18)(19,22,23,31,24,32,30,21)(20,26,27,25,28,29,33,34);(5,12)(10,17)(13,16)(14,18)(21,31)(22,32)(25,34)(26,29)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264562);