Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1946 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,12)(13,16)(14,15)(17,18,20,23)(19,21,24,22)(25,26,29,31)(27,28,32,30);(1,3)(2,7)(4,6)(5,8)(9,10,13,16)(11,12,15,14)(17,19)(18,21)(20,24)(22,23)(25,26,29,31)(27,28,32,30);(3,4)(5,7)(11,15)(12,14)(17,19,20,24)(18,22,23,21)(25,27,29,32)(26,30,31,28);(1,4,6,3)(2,7,8,5)(9,11,13,15)(10,14,16,12)(17,19)(18,22)(20,24)(21,23)(25,27)(26,28)(29,32)(30,31);(1,5,6,7)(2,3,8,4)(9,12,13,14)(10,15,16,11)(17,18)(19,22)(20,23)(21,24)(25,28)(26,27)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5135189);