Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6366542 |
| 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 | #18566 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1601 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)(3,6)(4,7)(5,8)(9,12)(10,14)(11,13)(15,16)(17,18,23,28)(19,29,31,32)(20,30,24,25)(21,27,22,26);(1,3)(2,6)(4,11)(5,10)(7,13)(8,14)(9,15)(12,16)(17,19,23,31)(18,24,28,20)(21,25,22,30)(26,32,27,29);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,20)(18,19)(21,32)(22,29)(23,24)(25,27)(26,30)(28,31);(2,8)(6,14)(7,9)(13,15)(17,20)(18,25)(19,27)(21,29)(22,32)(23,24)(26,31)(28,30);(3,10)(6,14)(11,16)(13,15)(17,21,23,22)(18,26,28,27)(19,25,31,30)(20,29,24,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=h,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,e^c=eh,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, 6273749);