Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,5)(3,8,4,7)(9,10,12,14)(11,13,15,16)(17,18,21,24)(19,22,20,23)(25,26)(27,28)(29,37)(30,31)(32,38)(33,36)(34,40)(35,39);(1,2,6,5)(3,7,4,8)(11,15)(13,16)(26,31)(27,33)(29,38)(35,40);(1,3,6,4)(2,7,5,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,22)(20,23)(21,24)(25,27,30,33)(26,28,31,36)(29,34,38,39)(32,35,37,40);(1,4,6,3)(2,8,5,7)(17,19,21,20)(18,22,24,23)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,39)(38,40);(1,5,6,2)(3,8,4,7)(9,10)(11,13)(12,14)(15,16)(17,20,21,19)(18,23,24,22)(25,29)(26,32)(27,34)(28,35)(30,38)(31,37)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=fgh,b^2=i,c^2=fi,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4242387);