Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | #16291 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(6,7)(9,10)(11,16)(12,13)(14,15)(17,18,22,21)(19,24,20,23)(25,26,29,32)(27,31,28,30);(1,2,5,4)(3,7,8,6)(9,11)(10,12)(13,14)(15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32);(2,4)(3,8)(10,15)(11,13)(17,19,22,20)(18,23,21,24)(25,26,29,32)(27,31,28,30);(1,3,5,8)(2,6,4,7)(9,12,14,16)(10,11,15,13)(17,20,22,19)(18,23,21,24)(25,28,29,27)(26,31,32,30);(1,4,5,2)(3,7,8,6)(9,13,14,11)(10,16,15,12)(17,21,22,18)(19,23,20,24)(25,26)(27,30)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=gi,d^2=f,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5114435);