Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10004 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10009 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,9,14)(5,15,6,17)(7,13,8,12)(10,18,11,16)(19,20,23,27)(21,25,29,34)(22,24,30,26)(28,33,31,32);(3,5)(4,10)(6,9)(7,18)(8,16)(11,14)(12,17)(13,15)(20,24)(22,30)(25,33)(26,27)(28,31)(32,34);(3,6,9,5)(4,11,14,10)(7,18,8,16)(12,17,13,15)(19,21,23,29)(20,25,27,34)(22,31,30,28)(24,32,26,33)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 142516);