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×#41416 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7182 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)(24,25)(26,33)(30,32)(34,38);(9,17)(11,21)(13,22)(15,18)(23,24,28,25)(26,29,33,31)(27,32,35,30)(34,37,38,36);(3,4,5,6)(7,9)(8,13)(10,15)(11,19)(12,17)(14,21)(16,22)(18,20)(23,25,28,24)(26,29,33,31)(27,32,35,30)(34,36,38,37);(7,10)(8,14)(9,18)(11,13)(12,20)(15,17)(16,19)(21,22)(23,26)(24,29)(25,31)(27,34)(28,33)(30,36)(32,37)(35,38);(7,11)(8,15)(9,19)(10,13)(12,21)(14,17)(16,18)(20,22)(23,27,28,35)(24,30,25,32)(26,34,33,38)(29,36,31,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547169);