Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2048 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(11,15)(14,16)(17,18,21,24)(19,20,23,22)(25,26)(27,31)(28,30)(29,32);(1,3)(2,6)(4,7)(5,8)(17,18,21,24)(19,20,23,22)(25,26,29,32)(27,30,28,31);(1,2,5,4)(3,6,8,7)(9,10,13,12)(11,14,15,16)(17,19,21,23)(18,22,24,20)(25,26)(27,30)(28,31)(29,32);(1,4,5,2)(3,7,8,6)(9,11)(10,14)(12,16)(13,15)(17,18)(19,22)(20,23)(21,24)(25,27,29,28)(26,30,32,31);(1,3)(2,6)(4,7)(5,8)(9,12,13,10)(11,16,15,14)(17,20,21,22)(18,23,24,19)(25,28,29,27)(26,31,32,30)Polycyclic presentation:
2^9|a^2=fh,b^2=h,c^2=fi,d^2=f,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 810922);