Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#46229 |
| 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 | |
6646 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)(9,15)(13,18)(16,21)(20,22)(24,27)(25,31)(30,34)(32,37);(8,12)(9,15)(10,17)(20,22)(23,24,28,27)(25,29,31,33)(26,34,35,30)(32,38,37,36);(3,4)(5,6)(7,8,11,12)(9,13,15,18)(10,14,17,19)(16,20,21,22)(23,25)(24,29)(26,32)(27,33)(28,31)(30,36)(34,38)(35,37);(7,9,11,15)(8,13,12,18)(10,16,17,21)(14,20,19,22)(23,26,28,35)(24,30,27,34)(25,32,31,37)(29,36,33,38);(7,10)(8,14)(9,16)(11,17)(12,19)(13,20)(15,21)(18,22)(23,27,28,24)(25,33,31,29)(26,34,35,30)(32,38,37,36)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7610115);