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×#52074 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6656 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)(9,15,17,13)(11,18,19,14)(16,20,22,21);(8,10)(13,15)(14,18)(20,21)(23,24,27,30)(25,31,33,28)(26,34,35,29)(32,36,38,37);(3,4)(5,6)(7,9,12,17)(8,13,10,15)(11,16,19,22)(14,20,18,21)(23,24)(25,31)(26,29)(27,30)(28,33)(32,37)(34,35)(36,38);(7,10,12,8)(9,15,17,13)(11,18,19,14)(16,21,22,20)(23,25,27,33)(24,28,30,31)(26,32,35,38)(29,36,34,37);(7,11)(8,14)(9,16)(10,18)(12,19)(13,20)(15,21)(17,22)(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, 7615366);