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×#18628 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3519 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3256 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,11)(6,14)(7,9)(8,13)(10,16)(12,18)(15,17)(19,20,24,27)(21,29,30,33)(22,28,23,26)(25,32,34,31);(3,5)(4,8)(6,15)(7,12)(9,17)(10,18)(11,14)(13,16)(19,21)(20,25)(22,32)(23,31)(24,30)(26,33)(27,34)(28,29);(20,26)(22,23)(25,33)(27,28)(29,34)(31,32);(4,9)(5,12)(10,14)(13,15)(20,27)(21,30)(26,28)(31,32)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6255400);