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 | |
1722 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)(10,12)(14,16)(17,18,21,24)(19,23,20,22)(25,26)(27,30)(28,32)(29,31);(1,2,4,6)(3,5,8,7)(11,15)(14,16)(17,18,21,24)(19,22,20,23)(25,27)(26,28)(29,32)(30,31);(1,3)(2,5)(4,8)(6,7)(9,10,13,12)(11,14,15,16)(17,19,21,20)(18,22,24,23)(25,27,29,32)(26,30,31,28);(1,3)(2,5)(4,8)(6,7)(9,11,13,15)(10,14,12,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,32)(29,31);(1,2)(3,7)(4,6)(5,8)(9,12,13,10)(11,16,15,14)(17,20,21,19)(18,23,24,22)(25,28,29,30)(26,27,31,32)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=fg,d^2=gi,e^2=fg,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, 3886910);