Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6360376 |
| 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 | |
1559 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,9)(3,7,5,14)(4,11,13,15)(8,12,16,10)(18,23)(20,21)(24,31)(25,30)(26,32)(27,28);(1,3,6,5)(2,7,9,14)(4,12,13,10)(8,11,16,15)(17,18,22,25)(19,26,29,31)(20,30,21,23)(24,27,32,28);(1,4,6,13)(2,8,9,16)(3,10,5,12)(7,15,14,11)(17,18)(19,24)(20,23)(21,30)(22,25)(26,28)(27,31)(29,32);(1,3,6,5)(2,7,9,14)(4,10,13,12)(8,15,16,11)(17,19,22,29)(18,24,25,32)(20,27,21,28)(23,31,30,26);(17,20,22,21)(18,23,25,30)(19,27,29,28)(24,31,32,26)Polycyclic presentation:
2^9|a^2=hi,b^2=i,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=ci,d^b=df,e^b=eh,f^b=fh,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, 6267825);