Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20158 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#20221 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1531 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,9,7)(5,12,15,18)(6,11,8,16)(10,13,17,14)(19,20,23,26)(21,27,28,34)(22,30,32,25)(24,31,33,29);(3,5,9,15)(4,10,7,17)(6,14,8,13)(11,18,16,12)(21,28)(24,33)(27,34)(29,31);(3,6,9,8)(4,11,7,16)(5,13,15,14)(10,12,17,18)(19,21,23,28)(20,24,26,33)(22,31,32,29)(25,34,30,27);(3,7,9,4)(5,12,15,18)(6,16,8,11)(10,13,17,14)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6266770);