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×#42174 |
| 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 | |
6770 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)(23,24,28,27)(25,31,33,29)(26,34,35,30)(32,36,38,37);(9,15)(11,19)(13,17)(14,18)(23,25,28,33)(24,29,27,31)(26,32,35,38)(30,36,34,37);(3,4,5,6)(7,9)(8,13)(10,17)(11,16)(12,15)(14,20)(18,22)(19,21)(23,25)(24,29)(26,32)(27,31)(28,33)(30,36)(34,37)(35,38);(7,10,12,8)(9,13,15,17)(11,18,19,14)(16,20,21,22)(23,26,28,35)(24,30,27,34)(25,32,33,38)(29,36,31,37);(7,11)(8,14)(9,16)(10,18)(12,19)(13,20)(15,21)(17,22)(23,27,28,24)(25,31,33,29)(26,34,35,30)(32,37,38,36)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547864);