Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10530 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#10533 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
940 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,9,14)(5,15,16,18)(6,10,17,11)(7,13,8,12)(19,20,23,27)(21,28,29,32)(22,24,30,26)(25,33,34,31);(3,5)(4,10)(6,8)(7,17)(9,16)(11,14)(12,18)(13,15)(20,24)(21,29)(22,30)(25,32)(26,27)(28,34);(3,6,7,16,9,17,8,5)(4,11,12,18,14,10,13,15)(19,21,30,33,23,29,22,31)(20,25,24,32,27,34,26,28)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 146181);