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×#41417 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6976 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)(7,8,12,16)(9,13,17,22)(10,14,20,19)(11,15,21,18)(23,24)(25,31)(26,29)(27,30)(28,33)(32,37)(34,36)(35,38);(9,17)(11,21)(13,22)(15,18)(23,24,27,30)(25,28,33,31)(26,29,34,36)(32,35,38,37);(3,4,5,6)(7,9)(8,13)(10,15)(11,19)(12,17)(14,21)(16,22)(18,20)(23,24)(25,28)(26,29)(27,30)(31,33)(32,35)(34,36)(37,38);(7,10)(8,14)(9,18)(11,13)(12,20)(15,17)(16,19)(21,22)(23,25)(24,28)(26,32)(27,33)(29,35)(30,31)(34,38)(36,37);(7,11)(8,15)(9,19)(10,13)(12,21)(14,17)(16,18)(20,22)(23,26,27,34)(24,29,30,36)(25,32,33,38)(28,35,31,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547189);