Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6349684 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #62835 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2780 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,7,12)(3,13,4,9)(5,11,6,10)(8,15,16,14)(17,18,23,27)(19,29,25,32)(20,31,24,30)(21,26,22,28);(1,3,7,4)(2,8,12,16)(5,14,6,15)(9,10,13,11)(17,19,23,25)(18,24,27,20)(21,30,22,31)(26,29,28,32);(1,4,7,3)(2,9,12,13)(5,14,6,15)(8,10,16,11)(17,20,23,24)(18,25,27,19)(21,29,22,32)(26,30,28,31);(1,5,7,6)(2,10,12,11)(3,14,4,15)(8,13,16,9)(17,21,23,22)(18,26,27,28)(19,30,25,31)(20,32,24,29);(1,5,7,6)(2,10,12,11)(3,15,4,14)(8,9,16,13)(18,27)(19,25)(26,28)(30,31)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=eh,f^a=fh,g^a=gi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6256943);