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 | |
1564 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,7,12)(3,9,13,14)(4,10,15,16)(5,11,6,8)(17,18)(19,22)(20,28)(21,25)(23,27)(24,29)(26,31)(30,32);(2,8)(4,14)(5,6)(9,15)(10,16)(11,12)(17,19,22,18)(20,26,30,24)(21,25,23,27)(28,29,32,31);(1,3,7,13)(2,9,12,14)(4,8,15,11)(5,16,6,10)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28);(1,4,7,15)(2,10,12,16)(3,8,13,11)(5,14,6,9)(17,21,22,23)(18,25,19,27)(20,29,30,31)(24,32,26,28);(1,5,7,6)(2,8,12,11)(3,10,13,16)(4,14,15,9)(18,19)(24,26)(25,27)(28,32)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,d^2=hi,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, 6275591);