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×#42647 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6960 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)(7,8,12,15)(9,13,16,11)(10,14,20,22)(17,21,18,19)(24,26)(25,31)(30,34)(32,37);(9,16)(11,13)(17,18)(19,21)(23,24,28,26)(25,29,31,33)(27,34,35,30)(32,38,37,36);(3,4)(5,6)(7,9)(8,13)(10,18)(11,15)(12,16)(14,19)(17,20)(21,22)(23,25)(24,29)(26,33)(27,32)(28,31)(30,36)(34,38)(35,37);(7,10)(8,14)(9,17)(11,19)(12,20)(13,21)(15,22)(16,18)(23,26,28,24)(25,29,31,33)(27,34,35,30)(32,36,37,38);(7,11,12,13)(8,9,15,16)(10,19,20,21)(14,17,22,18)(23,27,28,35)(24,30,26,34)(25,32,31,37)(29,36,33,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7548496);