Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6366542 |
| 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 | #61667 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,7,12)(3,13,14,9)(4,10,15,16)(5,11,6,8)(17,18)(19,22)(20,23)(21,24)(25,28)(26,30)(27,29)(31,32);(2,8)(4,9)(5,6)(10,16)(11,12)(13,15)(17,19)(18,22)(20,27)(21,26)(23,29)(24,30)(25,31)(28,32);(1,3,7,14)(2,9,12,13)(4,11,15,8)(5,16,6,10)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(1,4)(2,10)(3,11)(5,9)(6,13)(7,15)(8,14)(12,16)(18,24)(22,30)(23,25)(29,31);(1,5,7,6)(2,8,12,11)(3,10,14,16)(4,9,15,13)(19,26)(22,30)(27,32)(29,31)Polycyclic presentation:
2^9|a^2=h,c^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,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,f^d=fh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6273751);