Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6366710 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1557 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,8)(5,6)(9,16)(10,13)(11,12)(14,15)(18,23)(19,27)(20,21)(24,28)(25,26)(31,32);(1,2,7,12)(3,13,4,9)(5,8,6,11)(10,14,16,15)(17,18,22,26)(19,28,27,31)(20,25,21,23)(24,29,32,30);(1,3,7,4)(2,9,12,13)(5,15,6,14)(8,16,11,10)(17,18)(19,24)(20,23)(21,25)(22,26)(27,32)(28,30)(29,31);(1,4,7,3)(2,10,12,16)(5,15,6,14)(8,13,11,9)(17,19,22,27)(18,24,26,32)(20,29,21,30)(23,31,25,28);(1,5,7,6)(2,11,12,8)(3,14,4,15)(9,16,13,10)(17,20,22,21)(18,25,26,23)(19,29,27,30)(24,28,32,31)Polycyclic presentation:
2^9|b^2=h,c^2=hi,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,f^d=fh59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6273976);