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 | |
1520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,8)(4,7)(11,15)(14,16)(18,22)(23,24)(25,26,30,31)(27,34,32,28)(29,37,39,33)(35,36,40,38);(1,2,6,5)(3,7,8,4)(10,13)(14,16)(26,31)(27,32)(33,37)(35,40);(1,3,6,8)(2,7,5,4)(9,10)(11,14)(12,13)(15,16)(17,18,21,22)(19,23,20,24)(25,27,30,32)(26,28,31,34)(29,35,39,40)(33,36,37,38);(1,4)(2,3)(5,8)(6,7)(17,19,21,20)(18,23,22,24)(25,28)(26,32)(27,31)(29,38)(30,34)(33,35)(36,39)(37,40);(1,5,6,2)(3,4,8,7)(9,11,12,15)(10,14,13,16)(17,20,21,19)(18,24,22,23)(25,29,30,39)(26,33,31,37)(27,35,32,40)(28,36,34,38)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=g,d^2=gi,e^2=fgh,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, 4347261);