Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,13)(12,16)(14,15)(17,18,21,22)(19,23,24,20)(26,29)(30,32);(1,3,5,4)(2,6,7,8)(10,15)(11,12)(17,19,21,24)(18,20,22,23)(27,31)(30,32);(9,11,14,12)(10,16,15,13)(17,18)(19,20)(21,22)(23,24)(25,26)(27,30)(28,29)(31,32);(1,2)(3,6)(4,8)(5,7)(9,12,14,11)(10,16,15,13)(17,18)(19,23)(20,24)(21,22)(25,27,28,31)(26,30,29,32);(1,4,5,3)(2,6,7,8)(9,13,14,16)(10,12,15,11)(17,20,21,23)(18,19,22,24)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=h,d^2=gi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4802694);