Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #35831 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,5)(3,4)(9,10,13,15)(11,12,16,14)(17,18,22,23)(19,27,28,24)(20,29,31,25)(21,26,32,30);(1,2,6,5)(3,8,4,7)(9,11,13,16)(10,14,15,12)(18,23)(19,28)(25,29)(26,30);(1,3,6,4)(2,7,5,8)(9,11)(10,12)(13,16)(14,15)(18,23)(21,32)(24,27)(25,29);(1,4,6,3)(2,8,5,7)(9,12)(10,11)(13,14)(15,16)(17,19)(18,24)(20,30)(21,25)(22,28)(23,27)(26,31)(29,32);(1,5,6,2)(3,8,4,7)(17,20,22,31)(18,25,23,29)(19,26,28,30)(21,24,32,27);(17,21,22,32)(18,26,23,30)(19,25,28,29)(20,24,31,27)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=ghi,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8247074);