Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
| direct product | C2×#34455 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6638 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);(3,4,6,5)(8,12)(9,15)(14,19)(16,21)(23,24)(25,27)(26,30)(28,36)(29,32)(31,37)(33,38)(34,35);(8,12)(13,18)(14,19)(20,22)(23,25,29,26)(24,30,32,27)(28,33,37,34)(31,38,36,35);(3,5,6,4)(7,8)(9,13)(10,14)(11,12)(15,18)(16,20)(17,19)(21,22)(23,26,29,25)(24,30,32,27)(28,34,37,33)(31,38,36,35);(7,9,11,15)(8,13,12,18)(10,16,17,21)(14,20,19,22)(23,27,29,30)(24,25,32,26)(28,35,37,38)(31,33,36,34);(7,10,11,17)(8,14,12,19)(9,16,15,21)(13,20,18,22)(23,28)(24,31)(25,33)(26,34)(27,35)(29,37)(30,38)(32,36)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533230);