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×#41425 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#553 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7128 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)(9,17,18,13)(10,14,20,22)(11,15,21,19)(24,28)(25,31)(30,35)(32,37);(9,18)(11,21)(13,17)(15,19)(23,24,27,28)(25,29,31,34)(26,30,33,35)(32,36,37,38);(3,4,5,6)(7,9,12,18)(8,13,16,17)(10,19,20,15)(11,14,21,22)(23,24)(25,29)(26,30)(27,28)(31,34)(32,36)(33,35)(37,38);(7,10)(8,14)(9,15)(11,13)(12,20)(16,22)(17,21)(18,19)(23,25,27,31)(24,29,28,34)(26,32,33,37)(30,36,35,38);(7,11)(8,15)(9,14)(10,13)(12,21)(16,19)(17,20)(18,22)(23,26,27,33)(24,30,28,35)(25,32,31,37)(29,36,34,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547399);