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×#46822 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2688 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7064 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)(9,13)(10,18)(11,12)(14,19)(15,16)(17,22)(20,21)(23,24,28,29)(25,31,33,26)(27,35,36,30)(32,34,38,37);(8,12)(9,15)(10,19)(20,22)(24,29)(26,31)(30,35)(34,37);(3,4)(5,6)(7,8)(9,16)(10,14)(11,12)(13,15)(17,22)(18,19)(20,21)(23,25)(24,26)(27,32)(28,33)(29,31)(30,34)(35,37)(36,38);(7,9)(8,13)(10,17)(11,15)(12,16)(14,20)(18,22)(19,21)(23,26)(24,25)(27,34)(28,31)(29,33)(30,32)(35,38)(36,37);(7,10,11,19)(8,14,12,18)(9,17,15,21)(13,20,16,22)(23,27,28,36)(24,30,29,35)(25,32,33,38)(26,34,31,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7607980);