Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6363930 |
| 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 | |
1581 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)(8,10)(9,15)(11,14)(12,16)(18,23)(20,21)(24,31)(25,30)(26,32)(27,28);(1,2,6,10)(3,7,5,8)(4,12,13,15)(9,14,16,11)(17,18,22,25)(19,26,29,31)(20,30,21,23)(24,27,32,28);(1,3,6,5)(2,8,10,7)(4,11,13,14)(9,12,16,15)(17,18)(19,24)(20,23)(21,30)(22,25)(26,28)(27,31)(29,32);(1,4)(2,9)(3,11)(5,14)(6,13)(7,15)(8,12)(10,16)(17,19,22,29)(18,24,25,32)(20,27,21,28)(23,31,30,26);(1,3,6,5)(2,7,10,8)(4,11,13,14)(9,15,16,12)(17,20,22,21)(18,23,25,30)(19,27,29,28)(24,31,32,26)Polycyclic presentation:
2^9|b^2=h,c^2=hi,d^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ei,f^b=fh,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, 6271292);