Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6368846 |
| 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 | #62466 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1537 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,7,10)(3,9,4,5)(6,14,16,11)(8,13,15,12)(17,18,22,27)(19,23,20,25)(21,24,30,26)(28,31,29,32);(1,3,7,4)(2,8,10,15)(5,14,9,11)(6,13,16,12)(17,19,22,20)(18,23,27,25)(21,28,30,29)(24,31,26,32);(1,4,7,3)(2,9,10,5)(6,13,16,12)(8,11,15,14)(18,24)(21,30)(23,31)(25,32)(26,27)(28,29);(1,5,7,9)(2,3,10,4)(6,15,16,8)(11,13,14,12)(17,20,22,19)(18,25,27,23)(21,29,30,28)(24,32,26,31);(2,10)(3,4)(11,14)(12,13)(19,20)(23,25)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=h,d^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,e^d=ei,f^d=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6276174);