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 |
| prime radical | #61791 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,13)(4,14)(5,11)(6,10)(7,12)(8,15)(9,16)(17,18,24,29)(19,30,35,44)(20,26,38,21)(22,28,23,27)(25,33,45,34)(31,47,36,32)(37,40,41,42)(39,48,46,43);(1,3,7,8)(2,4,12,16)(5,14,6,9)(10,13,11,15)(17,19,24,35)(18,25,29,45)(20,36,38,31)(21,39,26,46)(22,34,23,33)(27,30,28,44)(32,42,47,40)(37,43,41,48);(1,4,7,16)(2,8,12,3)(5,15,6,13)(9,10,14,11)(17,20)(18,26)(19,31)(21,29)(22,41)(23,37)(24,38)(25,39)(27,40)(28,42)(30,47)(32,44)(33,48)(34,43)(35,36)(45,46);(1,4,7,16)(2,9,12,14)(3,11,8,10)(5,13,6,15)(17,21,24,26)(18,20,29,38)(19,32,35,47)(22,40,23,42)(25,43,45,48)(27,37,28,41)(30,31,44,36)(33,39,34,46);(1,5,7,6)(2,10,12,11)(3,14,8,9)(4,13,16,15)(17,22,24,23)(18,27,29,28)(19,33,35,34)(20,37,38,41)(21,40,26,42)(25,44,45,30)(31,43,36,48)(32,39,47,46)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=i,d^2=h,e^2=h,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, 6273997);