Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6366710 |
| 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 | |
1582 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)(3,7,12,15)(4,10,5,14)(8,13,16,11)(18,24)(19,26)(20,28)(21,22)(25,29)(27,32);(1,3)(2,7)(4,13)(5,11)(6,12)(8,14)(9,15)(10,16)(17,18)(19,23)(20,29)(21,26)(22,24)(25,31)(27,30)(28,32);(1,3)(2,8)(4,11)(5,13)(6,12)(7,14)(9,16)(10,15)(17,19,23,18)(20,27,28,25)(21,26,22,24)(29,31,32,30);(2,9)(7,15)(8,16)(10,14)(17,20,23,28)(18,25,19,27)(21,30,22,31)(24,32,26,29);(1,4,6,5)(2,10,9,14)(3,11,12,13)(7,8,15,16)(17,21,23,22)(18,26,19,24)(20,30,28,31)(25,29,27,32)Polycyclic presentation:
2^9|a^2=hi,c^2=h,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,f^d=fh59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6274041);