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 | |
1570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(3,4)(10,14)(11,15)(17,18,23,21)(19,24,22,20)(25,26)(27,30)(28,32)(29,31);(1,2,5,6)(3,8,4,7)(9,10,13,14)(11,12,15,16)(17,19,23,22)(18,24,21,20)(25,27,29,32)(26,30,31,28);(1,3,5,4)(2,7,6,8)(9,11,13,15)(10,12,14,16)(17,20)(18,22)(19,21)(23,24)(25,28)(26,27)(29,30)(31,32);(1,2)(3,7)(4,8)(5,6)(9,10)(11,12)(13,14)(15,16)(17,21,23,18)(19,24,22,20)(25,27)(26,28)(29,32)(30,31);(1,4,5,3)(2,7,6,8)(9,12)(10,15)(11,14)(13,16)(17,22,23,19)(18,24,21,20)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=f,d^2=gi,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, 4802857);