Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357620 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1813 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,7)(3,10,9,5)(4,8,12,14)(11,16,15,13)(17,18)(19,24)(20,32)(21,26)(22,36)(23,27)(25,34)(28,41)(29,38)(30,39)(31,45)(33,40)(35,46)(37,44)(42,48)(43,47);(1,3,5,2,6,9,10,7)(4,11,13,8,12,15,16,14)(17,19)(18,24)(20,28)(21,30)(22,31)(23,29)(25,37)(26,39)(27,38)(32,41)(33,43)(34,44)(35,42)(36,45)(40,47)(46,48);(2,7)(3,9)(8,14)(11,15)(17,20,21,33,23,35,22,34)(18,25,36,46,27,40,26,32)(19,28,30,43,29,42,31,44)(24,37,45,48,38,47,39,41);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(19,29)(24,38)(28,42)(30,31)(37,47)(39,45)(41,48)(43,44);(4,12)(8,14)(11,15)(13,16)(17,21,23,22)(18,26,27,36)(19,30,29,31)(20,33,35,34)(24,39,38,45)(25,32,40,46)(28,43,42,44)(37,41,47,48)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=ghi,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=di,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264859);