Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1806 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)(3,8)(4,7)(5,6)(9,10,12,14)(11,15,16,13)(17,18,21,24)(19,20,23,22);(2,6)(7,8)(9,10,12,14)(11,13,16,15)(19,23)(20,22)(26,30)(31,32);(1,3,5,4)(2,7,6,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,23)(18,22,24,20)(25,26,29,30)(27,31,28,32);(1,4,5,3)(2,8,6,7)(9,10)(11,13)(12,14)(15,16)(17,20,21,22)(18,23,24,19)(25,27,29,28)(26,31,30,32);(1,2)(3,8)(4,7)(5,6)(9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,23)(21,24)(25,28,29,27)(26,32,30,31)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=hi,d^2=hi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3941028);