Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19328 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#19341 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3169 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1756 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);(4,8)(6,9)(10,14)(11,15)(13,16)(17,18)(19,20)(21,27)(22,30)(23,25)(24,28)(26,32)(29,34)(31,33);(3,4,9,15,7,11,6,8)(5,10,16,18,12,17,13,14);(3,5)(4,10)(6,13)(7,12)(8,14)(9,16)(11,17)(15,18)(19,21,23,29)(20,24,25,33)(22,31,32,28)(26,34,30,27);(5,12)(10,17)(13,16)(14,18)(20,25)(24,33)(26,30)(27,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261206);