Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#50899 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2631 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6988 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)(9,16,17,13)(10,11,14,21)(18,19,20,22)(24,28)(26,34)(29,31)(32,38);(9,17)(10,14)(11,21)(13,16)(23,24,27,28)(25,29,33,31)(26,35,34,30)(32,37,38,36);(3,4,5,6)(7,9,12,17)(8,13,15,16)(10,20,14,18)(11,19,21,22)(23,24)(25,31)(26,30)(27,28)(29,33)(32,37)(34,35)(36,38);(7,10,12,14)(8,11,15,21)(9,18,17,20)(13,22,16,19)(23,25)(24,29)(26,32)(27,33)(28,31)(30,36)(34,38)(35,37);(7,11)(8,14)(9,19)(10,15)(12,21)(13,18)(16,20)(17,22)(23,26,27,34)(24,30,28,35)(25,32,33,38)(29,36,31,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613926);