Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6366374 |
| 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 | |
1554 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)(10,12)(11,15)(13,14)(17,18)(19,23)(20,26)(21,30)(22,25)(24,27)(28,31)(29,32);(1,2,7,11)(3,12,4,9)(5,8,6,15)(10,13,16,14)(17,19)(18,23)(20,29)(21,27)(22,28)(24,30)(25,31)(26,32);(1,3,7,4)(2,9,11,12)(5,14,6,13)(8,16,15,10)(17,19)(18,24)(20,27)(21,29)(22,28)(23,30)(25,32)(26,31);(1,4,7,3)(2,10,11,16)(5,14,6,13)(8,12,15,9)(18,25)(23,31)(24,32)(26,30);(1,5,7,6)(2,8,11,15)(3,13,4,14)(9,10,12,16)(17,20,22,21)(18,26,25,30)(19,27,28,29)(23,24,31,32)Polycyclic presentation:
2^9|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=di,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,f^d=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6273696);