Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1436 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,5)(3,8,4,7)(11,15)(14,16)(19,23)(21,24)(25,26)(27,33)(28,36)(29,30)(31,35)(32,38)(34,39)(37,40);(1,2,6,5)(3,7,4,8)(17,18,20,22)(19,21,23,24)(26,30)(31,33)(32,36)(39,40);(1,3,6,4)(2,7,5,8)(9,10,13,12)(11,14,15,16)(17,18)(19,21)(20,22)(23,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,38)(37,40);(1,4,6,3)(2,8,5,7)(9,11)(10,14)(12,16)(13,15)(25,27)(26,31)(28,37)(29,35)(30,33)(32,40)(34,38)(36,39);(1,5,6,2)(3,8,4,7)(9,12,13,10)(11,16,15,14)(17,19,20,23)(18,21,22,24)(25,28)(26,32)(27,34)(29,38)(30,36)(31,39)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=fi,d^2=g,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4359862);