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×#37545 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7002 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);(4,6)(7,8,11,12)(9,15,17,13)(10,14,18,19)(16,21,22,20)(26,34)(29,36)(32,38)(35,37);(8,12)(13,15)(14,19)(20,21)(23,24,27,30)(25,28,33,31)(26,29,34,36)(32,35,38,37);(3,4)(5,6)(7,8)(9,13)(10,14)(11,12)(15,17)(16,20)(18,19)(21,22)(23,24)(25,31)(26,29)(27,30)(28,33)(32,37)(34,36)(35,38);(7,9,11,17)(8,13,12,15)(10,16,18,22)(14,20,19,21)(23,25)(24,28)(26,32)(27,33)(29,35)(30,31)(34,38)(36,37);(7,10,11,18)(8,14,12,19)(9,16,17,22)(13,20,15,21)(23,26)(24,29)(25,32)(27,34)(28,35)(30,36)(31,37)(33,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541046);