Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3751 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,4)(5,9)(6,7)(8,12)(10,11)(13,14)(15,16)(17,18)(19,20)(21,25)(22,23)(24,28)(26,27)(29,30)(31,32)(33,34)(35,40)(36,37)(38,48)(39,42)(41,46)(43,47)(44,45);(2,7)(3,8)(9,16)(10,13)(18,23)(19,24)(25,30)(26,31)(33,35,39,38)(34,40,42,48)(36,43,46,45)(37,47,41,44);(2,7)(3,8)(9,16)(10,13)(17,19,22,24)(18,20,23,28)(21,26,29,31)(25,27,30,32)(33,36)(34,41)(35,43)(37,42)(38,45)(39,46)(40,44)(47,48);(1,3,6,8)(2,4,7,12)(5,13,15,10)(9,14,16,11)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,30)(29,32)(33,37,39,41)(34,36,42,46)(35,44,38,47)(40,45,48,43);(1,4)(2,8)(3,7)(5,14)(6,12)(9,10)(11,15)(13,16)(17,21,22,29)(18,25,23,30)(19,26,24,31)(20,27,28,32)(33,38,39,35)(34,40,42,48)(36,45,46,43)(37,47,41,44);(1,5,6,15)(2,9,7,16)(3,10,8,13)(4,11,12,14)Polycyclic presentation:
2^9|b^2=ghi,c^2=i,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8656928);