Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1991 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)(3,7)(4,6)(5,8)(10,14)(12,15)(19,23)(22,24)(25,26)(27,28)(29,30)(31,32);(1,3,6,5)(2,7,4,8)(10,14)(11,16)(18,21)(22,24)(26,30)(27,31);(1,2,6,4)(3,8,5,7)(9,10)(11,15)(12,16)(13,14)(17,18,20,21)(19,22,23,24)(25,27,29,31)(26,28,30,32);(1,4,6,2)(3,7,5,8)(9,11,13,16)(10,15,14,12);(1,5,6,3)(2,7,4,8)(9,12,13,15)(10,16,14,11)(17,19)(18,22)(20,23)(21,24)(25,28)(26,27)(29,32)(30,31)Polycyclic presentation:
2^9|b^2=i,c^2=ghi,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4557684);