Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7157001 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #44936 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1744 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,8,14,9,4,7,13)(3,15,30,10,20,16,19,21)(5,11,18,32,26,22,25,24)(6,27,17,23,29,31,28,12);(1,3)(2,10)(4,21)(5,6)(7,30)(8,19)(9,20)(11,23)(12,22)(13,16)(14,15)(17,25)(18,28)(24,31)(26,29)(27,32);(1,4)(2,9)(3,16,20,15)(5,24,26,32)(6,23)(7,13)(8,14)(10,19,21,30)(11,18,22,25)(12,29)(17,27)(28,31);(1,5,7,25,9,26,8,18)(2,11,13,24,4,22,14,32)(3,17,19,6,20,28,30,29)(10,31,15,23,21,27,16,12);(1,6,7,28,9,29,8,17)(2,12,14,27,4,23,13,31)(3,18,19,5,20,25,30,26)(10,32,16,22,21,24,15,11)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,d^2=f,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=eh,h^a=hi,c^b=ch,d^b=df,e^b=eh,f^b=fi,h^b=hi,d^c=dh,f^c=fi,g^c=gi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7064198);