Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10304 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#10676 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1016 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,6,8)(5,9,10,7)(19,20,22,26,23,27,31,24)(21,28,29,33,30,34,32,25);(3,5,6,10)(4,7,8,9)(11,12)(13,15)(14,17)(16,18)(20,24)(22,31)(25,28)(26,27)(29,32)(33,34);(12,15)(13,18)(16,17)(19,21,23,30)(20,25,27,33)(22,32,31,29)(24,28,26,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151136);