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 | |
1658 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)(3,6)(4,7)(5,8)(10,14)(11,15)(17,18)(19,20)(21,23)(22,24)(25,26,30,31)(27,34,35,32)(28,29,38,33)(36,39,40,37);(1,3)(2,5)(4,8)(6,7)(9,10,13,14)(11,12,15,16)(17,19,22,23)(18,21,24,20)(26,31)(27,35)(28,38)(37,39);(17,20,22,21)(18,23,24,19)(25,27,30,35)(26,32,31,34)(28,36,38,40)(29,37,33,39);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,15)(14,16)(17,21,22,20)(18,23,24,19)(25,28,30,38)(26,33,31,29)(27,36,35,40)(32,39,34,37);(1,2)(3,5)(4,7)(6,8)(9,12)(10,15)(11,14)(13,16)(17,18)(19,21)(20,23)(22,24)(25,29,30,33)(26,28,31,38)(27,37,35,39)(32,36,34,40)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=g,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, 4781188);