Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10378 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#10381 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #393 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #120 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
882 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);(3,4,6,9)(5,12)(7,11,8,16)(10,13)(14,17)(15,18)(19,20)(21,27)(22,30)(23,26)(24,28)(25,31)(29,34)(32,33);(4,9)(5,13,15,14)(10,18,17,12)(11,16)(22,31)(24,33)(25,30)(28,32);(3,5)(4,10)(6,15)(7,13)(8,14)(9,17)(11,18)(12,16)(19,21,23,29)(20,24,26,33)(22,32,31,28)(25,34,30,27)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 145509);