Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18618 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18655 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,8,12)(5,13,16,17)(6,10,7,11)(9,15,18,14)(20,25)(21,29)(22,23)(27,28)(30,31)(32,33);(3,5)(4,9)(6,14)(7,15)(8,16)(10,17)(11,13)(12,18)(19,20,24,28)(21,26,30,34)(22,25,23,27)(29,32,31,33);(3,6,8,7)(4,10,12,11)(5,14,16,15)(9,17,18,13)(19,21,24,30)(20,26,28,34)(22,29,23,31)(25,32,27,33);(3,6,8,7)(4,11,12,10)(5,15,16,14)(9,17,18,13)(19,22,24,23)(20,27,28,25)(21,29,30,31)(26,33,34,32)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6255429);