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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2040 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,4)(3,8,5,7)(9,10,13,12)(11,14,15,16)(17,18)(19,24)(20,21)(22,23)(25,26,29,30)(27,28,31,32);(1,3)(2,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(18,23)(21,24)(25,27,29,31)(26,28,30,32);(3,5)(7,8)(10,12)(14,16)(17,19,22,20)(18,21,23,24)(25,28)(26,27)(29,32)(30,31);(1,4,6,2)(3,7,5,8)(9,11,13,15)(10,14,12,16)(17,20,22,19)(18,24,23,21)(25,27)(26,28)(29,31)(30,32);(1,5,6,3)(2,8,4,7)(9,12,13,10)(11,14,15,16)(17,21,22,24)(18,20,23,19)(25,26)(27,28)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=i,d^2=f,e^2=f,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, 3850237);