Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6363930 |
| 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 | #61702 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1509 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,8)(3,10)(5,6)(9,16)(11,12)(13,14)(17,18)(19,27)(20,24)(21,35)(22,25)(23,30)(26,36)(28,43)(29,39)(31,38)(32,46)(33,47)(34,41)(37,45)(40,44)(42,48);(1,2)(3,7)(4,12)(5,10)(6,8)(9,14)(11,13)(15,16)(17,19,22,31)(18,23,25,39)(20,32,34,28)(21,29,36,30)(24,40,41,37)(26,38,35,27)(33,45,48,44)(42,43,47,46);(1,3,7,2)(4,11,15,9)(5,10,6,8)(12,14,16,13)(17,20,22,34)(18,24,25,41)(19,28,31,32)(21,33,36,48)(23,37,39,40)(26,42,35,47)(27,43,38,46)(29,44,30,45);(1,4)(2,9)(3,11)(5,13)(6,14)(7,15)(8,16)(10,12)(19,29)(21,36)(23,38)(26,35)(27,39)(28,44)(30,31)(32,45)(33,48)(37,46)(40,43)(42,47);(1,5,7,6)(2,8,3,10)(4,13,15,14)(9,16,11,12)(18,25)(23,39)(24,41)(26,35)(27,38)(37,40)(42,47)(43,46)Polycyclic presentation:
2^9|b^2=hi,c^2=h,e^2=i,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=ei,f^b=fh,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, 6271388);