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 |
| prime radical | #61788 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,7,11)(3,4,9,14)(5,10,6,8)(12,16,15,13)(17,18)(19,29)(20,25)(21,30)(22,27)(23,26)(24,28)(31,32);(2,8)(3,9)(4,13)(5,6)(10,11)(14,16)(17,19)(18,21)(20,28)(22,30)(23,25)(24,31)(26,29)(27,32);(1,3,7,9)(2,4,11,14)(5,15,6,12)(8,16,10,13)(17,20)(18,19)(21,24)(22,29)(23,32)(25,26)(27,30)(28,31);(1,4)(2,9)(3,11)(5,13)(6,16)(7,14)(8,15)(10,12)(17,21)(18,25)(19,26)(20,24)(22,29)(23,32)(27,31)(28,30);(1,5,7,6)(2,8,11,10)(3,12,9,15)(4,13,14,16)(17,22,24,23)(18,26,28,27)(19,30,31,25)(20,32,21,29)Polycyclic presentation:
2^9|a^2=hi,c^2=hi,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, 6273991);