Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6367198 |
| 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 | #18510 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,7)(4,5)(8,10)(9,15)(11,16)(12,14)(17,18,24,29)(19,30,25,32)(20,31,21,26)(22,28,23,27);(1,2,6,8)(3,11,13,15)(4,7,5,10)(9,14,16,12)(17,19,24,25)(18,20,29,21)(22,31,23,26)(27,32,28,30);(2,8)(7,10)(9,16)(11,15)(17,20,24,21)(18,25,29,19)(22,30,23,32)(26,27,31,28);(1,3,6,13)(2,9,8,16)(4,12,5,14)(7,15,10,11)(17,21,24,20)(18,26,29,31)(19,28,25,27)(22,30,23,32);(1,4,6,5)(2,10,8,7)(3,12,13,14)(9,11,16,15)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=i,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6274673);