Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6368046 |
| 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 | |
1558 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)(3,9)(4,10)(5,11)(6,8)(7,12)(13,16)(14,15)(17,18,22,19)(20,27,29,32)(21,26,31,24)(23,30,25,28);(2,8)(4,15)(5,6)(9,16)(10,13)(11,12)(17,19,22,18)(20,25,29,23)(21,26,31,24)(27,28,32,30);(1,3)(2,9)(4,11)(5,10)(6,13)(7,14)(8,16)(12,15)(17,20,22,29)(18,23,19,25)(21,30,31,28)(24,32,26,27);(1,4,7,16)(2,10,12,13)(3,11,14,8)(5,15,6,9);(1,5,7,6)(2,8,12,11)(3,13,14,10)(4,15,16,9)(18,19)(23,25)(24,26)(27,32)Polycyclic presentation:
2^9|a^2=i,b^2=i,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,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,f^d=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6275490);