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 | |
1950 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)(4,5)(6,7)(9,10,13,12)(11,15,16,14)(17,18,22,20)(19,21,24,23)(26,29)(30,32);(1,3,6,4)(2,5,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,24)(18,23,20,21)(27,31)(30,32);(2,7)(5,8)(10,12)(14,15)(17,18,22,20)(19,21,24,23)(25,26)(27,30)(28,29)(31,32);(1,4,6,3)(2,8,7,5)(9,11,13,16)(10,14,12,15)(17,20,22,18)(19,23,24,21)(25,27)(26,30)(28,31)(29,32);(1,5,6,8)(2,4,7,3)(9,12,13,10)(11,14,16,15)(17,21,22,23)(18,19,20,24)(25,26)(27,30)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=fi,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, 3850239);