Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6361928 |
| 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 | #62368 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1524 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)(3,7)(4,11)(5,14)(6,9)(8,10)(12,16)(13,15)(17,18,22,25)(19,27,31,43)(20,24,34,41)(21,35,38,26)(23,36,28,30)(29,45,47,40)(32,46,48,39)(33,44,37,42);(1,3,6,5)(2,7,9,14)(4,12,13,10)(8,11,16,15)(17,19,22,31)(18,23,25,28)(20,32,34,48)(21,36,38,30)(24,40,41,45)(26,43,35,27)(29,37,47,33)(39,44,46,42);(1,4)(2,8)(3,10)(5,12)(6,13)(7,15)(9,16)(11,14)(17,18)(19,28)(20,24)(21,26)(22,25)(23,31)(27,36)(29,46)(30,43)(32,45)(33,42)(34,41)(35,38)(37,44)(39,47)(40,48);(2,9)(7,14)(8,16)(11,15)(17,20,22,34)(18,24,25,41)(19,29,31,47)(21,37,38,33)(23,39,28,46)(26,44,35,42)(27,45,43,40)(30,48,36,32);(1,3,6,5)(2,7,9,14)(4,10,13,12)(8,15,16,11)(18,25)(23,28)(24,41)(26,35)(27,43)(39,46)(40,45)(42,44)Polycyclic presentation:
2^9|a^2=h,b^2=hi,d^2=h,e^2=i,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=chi,d^b=df,f^b=fh,e^c=eh,g^c=gi,f^d=fh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6269212);