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×#52073 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6928 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)(9,14)(10,15)(11,19)(12,13)(16,20)(17,18)(21,22);(8,13)(10,14)(16,19)(18,21)(23,24,27,30)(25,31,33,28)(26,34,35,29)(32,36,38,37);(3,4)(5,6)(7,9,12,15)(8,14,13,10)(11,17,20,22)(16,21,19,18)(23,24)(25,31)(26,29)(27,30)(28,33)(32,37)(34,35)(36,38);(7,10,12,14)(8,15,13,9)(11,18,20,21)(16,22,19,17)(23,25,27,33)(24,28,30,31)(26,32,35,38)(29,36,34,37);(7,11)(8,16)(9,17)(10,18)(12,20)(13,19)(14,21)(15,22)(23,26,27,35)(24,29,30,34)(25,32,33,38)(28,36,31,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7615362);