Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6367198 |
| 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 | |
1695 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,10)(3,8,5,7)(4,9,12,16)(11,15,13,14)(19,26)(20,21)(23,30)(24,28)(27,29)(31,32);(2,7)(3,5)(4,11)(8,10)(9,16)(12,13)(17,18,22,25)(19,23,27,32)(20,28,21,24)(26,30,29,31);(1,3,6,5)(2,8,10,7)(4,11,12,13)(9,15,16,14)(17,19)(18,23)(20,26)(21,29)(22,27)(24,31)(25,32)(28,30);(1,4,6,12)(2,9,10,16)(3,11,5,13)(7,14,8,15);(1,3,6,5)(2,7,10,8)(4,11,12,13)(9,14,16,15)(17,20,22,21)(18,24,25,28)(19,26,27,29)(23,31,32,30)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,d^2=h,e^2=hi,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,e^b=ehi,f^b=fh,g^c=gi,f^d=fh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72339751, #72339760, #72339778, … |
| maximal normal subgroup of | #72339751, #72339760, #72339778, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6274637);