Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19733 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#19738 |
| 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 | |
1934 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);(5,11)(6,13)(8,14)(10,16)(12,17)(15,18)(19,20,24,27)(21,28,26,23)(22,30,31,29)(25,33,34,32);(3,4)(5,8)(6,10)(7,9)(11,14)(12,15)(13,16)(17,18)(19,21,23,20,24,26,28,27)(22,29,32,25,31,30,33,34);(3,5,6,11,7,12,13,17)(4,8,10,14,9,15,16,18)(19,22,28,33,24,31,23,32)(20,25,21,29,27,34,26,30);(4,9)(8,15)(10,16)(14,18)(22,31)(25,34)(29,30)(32,33)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264576);