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 | |
1492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,5,8,7)(10,14)(11,12)(19,23)(22,24)(25,26)(27,31)(28,32)(29,30);(1,2)(3,5)(4,6)(7,8)(9,10,13,14)(11,16,12,15)(18,21)(22,24)(25,27,30,28)(26,31,29,32);(9,11,13,12)(10,15,14,16)(17,18)(19,22)(20,21)(23,24)(25,28,30,27)(26,31,29,32);(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(25,29,30,26)(27,31,28,32);(1,3,4,8)(2,5,6,7)(9,12,13,11)(10,15,14,16)(17,19)(18,22)(20,23)(21,24)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=g,d^2=fi,e^2=f,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, 4801143);