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 | |
4157 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,9,11,4)(5,13,14,8)(10,12,16,15)(17,18)(19,20)(21,25)(22,30)(23,24)(26,31)(27,32)(28,29)(33,34,39,38)(35,41,36,40)(37,42,46,45)(43,48,44,47);(2,7)(4,9)(8,13)(12,15)(17,19)(18,20)(21,23)(22,27)(24,25)(26,28)(29,31)(30,32)(33,35,39,36)(34,40,38,41)(37,43,46,44)(42,47,45,48);(2,7)(4,9)(5,14)(10,16)(18,24)(19,21)(22,31)(28,32)(33,35,39,36)(34,41,38,40)(37,44,46,43)(42,47,45,48);(1,3,6,11)(2,4,7,9)(5,10,14,16)(8,12,13,15)(17,20)(18,19)(21,24)(22,28)(23,25)(26,27)(29,30)(31,32)(33,36,39,35)(34,41,38,40)(37,43,46,44)(42,47,45,48);(1,4)(2,3)(5,12)(6,9)(7,11)(8,10)(13,16)(14,15)(17,21,23,19)(18,25,24,20)(22,29,31,27)(26,32,30,28)(33,37,39,46)(34,42,38,45)(35,43,36,44)(40,47,41,48);(1,5)(2,8)(3,10)(4,12)(6,14)(7,13)(9,15)(11,16)(17,22,23,31)(18,26,24,30)(19,27,21,29)(20,28,25,32)(33,38,39,34)(35,41,36,40)(37,45,46,42)(43,48,44,47)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8853422);