Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574270 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1416 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,18,23,20)(19,26,28,32)(21,25,22,30)(24,31,29,27);(2,6)(7,8)(10,14)(11,12)(17,19,22,27,23,28,21,31)(18,24,25,26,20,29,30,32);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,19,21,31,23,28,22,27)(18,24,30,32,20,29,25,26);(1,4,5,3)(2,8,6,7)(11,12)(15,16)(17,20,23,18)(19,24,28,29)(21,25,22,30)(26,31,32,27);(9,12,13,11)(10,16,14,15)(17,21,23,22)(18,25,20,30)(19,27,28,31)(24,32,29,26)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=fi,d^2=fghi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^c=ei,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6532014);