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×#19714 |
| 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 | |
1990 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);(3,4,8,11)(5,12,10,7)(6,9,14,16)(13,18,17,15)(19,20,25,30)(21,31,32,27)(22,34,33,26)(23,29,24,28);(3,5,7,4,8,10,12,11)(6,13,15,9,14,17,18,16)(19,21,23,33,25,32,24,22)(20,26,28,31,30,34,29,27);(3,6)(4,9)(5,13)(7,15)(8,14)(10,17)(11,16)(12,18)(19,22,24,32,25,33,23,21)(20,27,29,34,30,31,28,26);(6,14)(9,16)(13,17)(15,18)(21,32)(22,33)(26,34)(27,31)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264546);