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×#42171 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6792 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)(7,8,12,10)(9,13,15,17)(11,14,19,18)(16,20,21,22)(23,24)(25,31)(26,29)(27,30)(28,33)(32,37)(34,36)(35,38);(9,15)(11,19)(13,17)(14,18)(23,24,27,30)(25,28,33,31)(26,29,34,36)(32,35,38,37);(3,4,5,6)(7,9)(8,13)(10,17)(11,16)(12,15)(14,20)(18,22)(19,21)(23,24)(25,28)(26,29)(27,30)(31,33)(32,35)(34,36)(37,38);(7,10,12,8)(9,13,15,17)(11,18,19,14)(16,20,21,22)(23,25,27,33)(24,28,30,31)(26,32,34,38)(29,35,36,37);(7,11)(8,14)(9,16)(10,18)(12,19)(13,20)(15,21)(17,22)(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, 7547862);