Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6365322 |
| 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 | #61660 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,9)(3,6,10,14)(4,7,12,8)(11,13,16,15)(17,18,22,25)(19,27,30,43)(20,24,34,41)(21,35,38,26)(23,36,28,31)(29,45,47,40)(32,46,48,39)(33,44,37,42);(1,3)(2,6)(4,11)(5,10)(7,13)(8,15)(9,14)(12,16)(17,19)(18,23)(20,32)(21,36)(22,30)(24,40)(25,28)(26,43)(27,35)(29,33)(31,38)(34,48)(37,47)(39,42)(41,45)(44,46);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,18)(19,28)(20,24)(21,26)(22,25)(23,30)(27,36)(29,46)(31,43)(32,45)(33,42)(34,41)(35,38)(37,44)(39,47)(40,48);(17,20,22,34)(18,24,25,41)(19,29,30,47)(21,37,38,33)(23,39,28,46)(26,44,35,42)(27,45,43,40)(31,48,36,32);(3,10)(6,14)(11,16)(13,15)(18,25)(19,30)(24,41)(26,35)(29,47)(31,36)(32,48)(42,44)Polycyclic presentation:
2^9|a^2=hi,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,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=fh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6272541);