Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#42801 |
| 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 | |
6592 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)(24,26)(25,31)(30,34)(32,37);(7,8,11,14)(9,12,17,15)(10,18,19,13)(16,21,22,20)(23,24,28,26)(25,29,31,33)(27,34,35,30)(32,38,37,36);(3,4)(5,6)(7,8)(9,15)(10,13)(11,14)(12,17)(16,21)(18,19)(20,22)(23,25)(24,29)(26,33)(27,32)(28,31)(30,36)(34,38)(35,37);(7,9)(8,12)(10,16)(11,17)(13,20)(14,15)(18,21)(19,22)(23,26,28,24)(25,29,31,33)(27,34,35,30)(32,36,37,38);(7,10,11,19)(8,13,14,18)(9,16,17,22)(12,20,15,21)(23,27,28,35)(24,30,26,34)(25,32,31,37)(29,36,33,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7548648);