Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6364762 |
| 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 | #62378 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,10)(3,7,12,16)(4,9,5,14)(8,13,15,11)(17,18)(19,27)(20,24)(21,34)(22,25)(23,29)(26,37)(28,45)(30,43)(31,48)(32,44)(33,41)(35,39)(36,42)(38,47)(40,46);(1,3)(2,7)(4,13)(5,11)(6,12)(8,14)(9,15)(10,16)(17,19,22,30)(18,23,25,39)(20,31,33,47)(21,35,37,29)(24,40,41,45)(26,43,34,27)(28,36,46,32)(38,44,48,42);(1,3)(2,8)(4,11)(5,13)(6,12)(7,14)(9,16)(10,15)(17,18)(19,23)(20,24)(21,26)(22,25)(27,29)(28,38)(30,39)(31,40)(32,42)(33,41)(34,37)(35,43)(36,44)(45,47)(46,48);(17,20,22,33)(18,24,25,41)(19,28,30,46)(21,36,37,32)(23,38,39,48)(26,44,34,42)(27,45,43,40)(29,47,35,31);(1,4,6,5)(2,9,10,14)(3,11,12,13)(7,8,16,15)(18,25)(23,39)(24,41)(26,34)(27,43)(38,48)(40,45)(42,44)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^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,d^b=df,e^b=ei,f^b=fh,e^c=eh,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, 6272100);