Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1827 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)(9,10,12,14)(11,15,16,13)(17,18,23,21)(19,24,22,20)(25,26,29,32)(27,31,28,30);(1,3,6,5)(2,7,4,8)(11,16)(13,15)(17,19,23,22)(18,24,21,20)(25,27,29,28)(26,30,32,31);(1,2,6,4)(3,8,5,7)(9,10)(11,13)(12,14)(15,16)(17,20)(18,22)(19,21)(23,24)(25,26)(27,31)(28,30)(29,32);(1,4,6,2)(3,7,5,8)(9,11)(10,13)(12,16)(14,15)(17,21,23,18)(19,24,22,20);(1,5,6,3)(2,7,4,8)(9,10)(11,13)(12,14)(15,16)(17,22,23,19)(18,24,21,20)(25,28,29,27)(26,31,32,30)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=ghi,d^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2161073);