Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#37635 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6806 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 38 points:
(1,2);(3,4,5,6)(7,8,12,9)(10,15,18,13)(11,14,19,16)(17,21,22,20)(27,35)(30,34)(32,38)(36,37);(8,9)(13,15)(14,16)(20,21)(23,24,28,26)(25,31,33,29)(27,30,35,34)(32,37,38,36);(3,4)(5,6)(7,9,12,8)(10,15,18,13)(11,16,19,14)(17,21,22,20)(23,25,28,33)(24,29,26,31)(27,32,35,38)(30,36,34,37);(7,10,12,18)(8,13,9,15)(11,17,19,22)(14,20,16,21)(23,26,28,24)(25,29,33,31)(27,34,35,30)(32,36,38,37);(7,11,12,19)(8,14,9,16)(10,17,18,22)(13,20,15,21)(23,27)(24,30)(25,32)(26,34)(28,35)(29,36)(31,37)(33,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541119);