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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4177 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,6,4)(5,9,15,16)(11,13,14,12)(17,18)(19,21)(20,29)(22,30)(23,25)(24,28)(26,27)(31,32)(33,34,39,43)(35,37,44,40)(36,41,46,48)(38,42,47,45);(1,3,7,6)(2,8,10,4)(5,11,15,14)(9,13,16,12)(17,19,23,28)(18,24,25,21)(20,26,30,32)(22,27,29,31)(33,35)(34,40)(36,38)(37,43)(39,44)(41,45)(42,48)(46,47);(1,4,7,8)(2,3,10,6)(5,13,15,12)(9,14,16,11)(17,20,23,30)(18,22,25,29)(19,26,28,32)(21,31,24,27)(33,36,39,46)(34,41,43,48)(35,38,44,47)(37,42,40,45);(1,4)(2,3)(5,13)(6,10)(7,8)(9,14)(11,16)(12,15)(17,20)(18,22)(19,26)(21,31)(23,30)(24,27)(25,29)(28,32)(33,37,39,40)(34,35,43,44)(36,45,46,42)(38,41,47,48);(1,5)(2,9)(3,11)(4,12)(6,14)(7,15)(8,13)(10,16)(17,21,23,24)(18,19,25,28)(20,27,30,31)(22,32,29,26)(33,36)(34,41)(35,38)(37,42)(39,46)(40,45)(43,48)(44,47);(1,6,7,3)(2,4,10,8)(5,14,15,11)(9,12,16,13)(17,22)(18,20)(19,27)(21,32)(23,29)(24,26)(25,30)(28,31)(33,38)(34,42)(35,36)(37,41)(39,47)(40,48)(43,45)(44,46)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=gi,d^2=gh,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8937049);