Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19621 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#19628 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2052 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);(3,4,7,9)(5,11,12,18)(6,14,15,10)(8,16,17,13)(19,20)(21,27)(22,24)(23,31)(25,29)(26,30)(28,33)(32,34);(3,5,6,13,7,12,15,16)(4,8,10,11,9,17,14,18)(20,24)(25,32)(26,31)(27,33);(4,9)(8,17)(10,14)(11,18)(19,21,23,29,22,28,30,34)(20,25,26,27,24,32,31,33);(5,12)(8,17)(11,18)(13,16)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264304);