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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1420 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)(3,7)(4,6)(5,8)(9,10,13,12)(11,15,16,14)(17,18,22,24)(19,20,23,21)(25,26)(27,34)(28,33)(29,38)(30,31)(32,35)(36,37)(39,40);(1,3,6,5)(2,7,4,8)(10,12)(14,15)(17,19)(18,21)(20,24)(22,23)(26,31)(27,35)(28,38)(37,39);(1,2,6,4)(3,8,5,7)(9,11)(10,14)(12,15)(13,16)(17,20,22,21)(18,23,24,19)(25,27)(26,32)(28,36)(29,37)(30,35)(31,34)(33,39)(38,40);(1,4,6,2)(3,7,5,8)(9,12,13,10)(11,15,16,14)(17,19)(18,20)(21,24)(22,23)(25,28,30,38)(26,33,31,29)(27,36,35,40)(32,39,34,37);(1,5,6,3)(2,7,4,8)(9,11)(10,15)(12,14)(13,16)(17,21,22,20)(18,23,24,19)(25,29,30,33)(26,28,31,38)(27,37,35,39)(32,36,34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=fg,d^2=g,e^2=h,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, 4781174);