Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6365694 |
| 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 | |
1551 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,7)(3,5)(8,15)(9,10)(11,14)(12,16)(17,18,21,24)(19,26,28,32)(20,25,22,29)(23,30,31,27);(1,2)(3,7)(4,12)(5,9)(6,10)(8,11)(13,15)(14,16)(17,18)(19,23)(20,29)(21,24)(22,25)(26,27)(28,31)(30,32);(1,3,6,5)(2,7,10,9)(4,11,13,14)(8,15,16,12)(17,19,21,28)(18,23,24,31)(20,27,22,30)(25,32,29,26);(1,4,6,13)(2,8,10,16)(3,11,5,14)(7,15,9,12)(18,24)(23,31)(25,29)(26,32);(1,3,6,5)(2,9,10,7)(4,11,13,14)(8,12,16,15)(17,20,21,22)(18,25,24,29)(19,27,28,30)(23,32,31,26)Polycyclic presentation:
2^9|a^2=i,c^2=hi,d^2=h,e^2=hi,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,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, 6273334);