Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10104 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10115 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #121 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1264 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)(5,12)(6,8)(7,17)(9,11)(10,13)(14,18)(15,16)(19,20,23,26)(21,27,28,34)(22,30,31,25)(24,32,33,29);(3,5,8,14)(4,10,6,16)(7,15,9,13)(11,18,17,12)(22,31)(25,30)(27,34)(29,32);(3,6,8,4)(5,12,14,18)(7,17,9,11)(10,13,16,15)(19,21)(20,24)(22,32)(23,28)(25,34)(26,33)(27,30)(29,31)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 143623);