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 | |
4162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,9,10,7)(4,8,13,14)(11,16,12,15)(34,39)(36,40)(41,47)(44,45);(2,6)(3,10)(8,14)(11,12)(17,18,22,23)(19,24,28,30)(20,27,29,25)(21,26,31,32)(33,34,38,39)(35,40,43,36)(37,41,46,47)(42,45,48,44);(2,6)(7,9)(8,14)(15,16)(18,23)(20,29)(24,30)(26,32)(33,35,38,43)(34,40,39,36)(37,42,46,48)(41,45,47,44);(1,3)(2,7)(4,12)(5,10)(6,9)(8,16)(11,13)(14,15)(17,19)(18,24)(20,26)(21,25)(22,28)(23,30)(27,31)(29,32)(33,36)(34,35)(37,45)(38,40)(39,43)(41,48)(42,47)(44,46);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,20,22,29)(18,25,23,27)(19,26,28,32)(21,30,31,24)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,47)(46,48);(1,4,5,13)(2,8,6,14)(3,11,10,12)(7,15,9,16)(17,21)(18,26)(19,27)(20,30)(22,31)(23,32)(24,29)(25,28)(33,37,38,46)(34,41,39,47)(35,42,43,48)(36,44,40,45)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8525330);