Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#46828 |
| 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 | |
6864 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)(7,8,12,14)(9,15,16,11)(10,13,20,22)(17,21,18,19)(24,28)(26,34)(29,31)(32,38);(9,16)(11,15)(17,18)(19,21)(23,24,27,28)(25,31,33,29)(26,35,34,30)(32,36,38,37);(3,4)(5,6)(7,9)(8,11)(10,18)(12,16)(13,21)(14,15)(17,20)(19,22)(23,24)(25,31)(26,30)(27,28)(29,33)(32,37)(34,35)(36,38);(7,10)(8,13)(9,17)(11,19)(12,20)(14,22)(15,21)(16,18)(23,25,27,33)(24,29,28,31)(26,32,34,38)(30,36,35,37);(7,11)(8,9)(10,19)(12,15)(13,17)(14,16)(18,22)(20,21)(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, 7607919);