Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1435 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,7)(3,4)(11,15)(14,16)(17,18,20,22)(19,23,24,21)(26,30)(28,35)(31,36)(34,39);(1,2)(3,8)(4,5)(6,7)(9,10,13,12)(11,14,15,16)(19,24)(21,23)(26,30)(27,33)(32,37)(34,39);(1,3,6,4)(2,8,7,5)(9,10,13,12)(11,14,15,16)(17,18)(19,21)(20,22)(23,24)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,4,6,3)(2,8,7,5)(9,11)(10,14)(12,16)(13,15)(17,19,20,24)(18,21,22,23)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(1,5,6,8)(2,4,7,3)(9,12,13,10)(11,16,15,14)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)Polycyclic presentation:
2^9|a^2=fgh,b^2=ghi,c^2=fghi,d^2=gh,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4835937);