Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6363286 |
| 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 | #61495 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1548 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,7,12)(3,9,14,4)(5,11,6,10)(8,15,16,13)(17,18)(19,25)(20,28)(21,23)(22,26)(24,30)(27,32)(29,31);(1,3,7,14)(2,8,12,16)(4,10,9,11)(5,13,6,15);(1,3)(2,9)(4,12)(5,15)(6,13)(7,14)(8,10)(11,16)(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25);(1,4,7,9)(2,3,12,14)(5,8,6,16)(10,13,11,15);(1,5,7,6)(2,10,12,11)(3,13,14,15)(4,8,9,16)(18,23)(22,31)(24,28)(25,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=i,d^2=h,e^2=h,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,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, 6270529);