Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| 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 | |
1546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(4,7)(9,10,13,14)(11,15,16,12)(17,18)(19,24)(20,21)(22,23)(25,26,30,31)(27,28,35,32)(29,33,38,39)(34,36,37,40);(1,2)(3,4)(5,6)(7,8)(9,11,13,16)(10,12,14,15)(18,23)(19,20)(26,31)(28,32)(33,39)(36,40);(2,6)(4,7)(10,14)(11,16)(17,19,22,20)(18,24,23,21)(25,27)(26,32)(28,31)(29,37)(30,35)(33,36)(34,38)(39,40);(1,3)(2,7)(4,6)(5,8)(9,12)(10,11)(13,15)(14,16)(17,20,22,19)(18,24,23,21)(25,28,30,32)(26,27,31,35)(29,36,38,40)(33,34,39,37);(1,4,5,7)(2,3,6,8)(9,10)(11,12)(13,14)(15,16)(17,21,22,24)(18,19,23,20)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,40)(35,37)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=fhi,d^2=i,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4304108);