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×#37607 |
| 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 | |
7046 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)(10,18)(14,19)(16,22)(20,21)(24,25)(26,33)(27,35)(36,37);(8,12)(13,15)(14,19)(20,21)(23,24,28,25)(26,29,33,31)(27,30,35,32)(34,36,38,37);(3,4)(5,6)(7,8)(9,15)(10,14)(11,12)(13,17)(16,21)(18,19)(20,22)(23,25,28,24)(26,29,33,31)(27,32,35,30)(34,36,38,37);(7,9,11,17)(8,13,12,15)(10,16,18,22)(14,20,19,21)(23,26,28,33)(24,29,25,31)(27,34,35,38)(30,36,32,37);(7,10)(8,14)(9,16)(11,18)(12,19)(13,20)(15,21)(17,22)(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, 7541069);