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 | |
4478 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)(3,8)(4,7)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18,22,23)(19,27,30,24)(20,25,31,32)(21,34,36,26)(28,41,43,37)(29,38,46,42)(33,40,35,39)(44,47,45,48);(1,3,5,7)(2,4,6,8)(10,13)(14,15)(18,23)(20,31)(24,27)(26,34)(28,43)(33,35)(38,42)(44,45);(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,24,23,27)(20,28,31,43)(21,29,36,46)(25,37,32,41)(26,38,34,42)(33,44,35,45)(39,47,40,48);(2,6)(3,7)(9,11,12,16)(10,14,13,15)(17,18)(19,27)(20,32)(21,26)(22,23)(24,30)(25,31)(28,37)(29,42)(33,40)(34,36)(35,39)(38,46)(41,43)(44,47)(45,48);(1,2)(3,4)(5,6)(7,8)(17,20)(18,25)(19,28)(21,35)(22,31)(23,32)(24,37)(26,40)(27,41)(29,45)(30,43)(33,36)(34,39)(38,48)(42,47)(44,46);(1,4)(2,7)(3,6)(5,8)(17,21,22,36)(18,26,23,34)(19,29,30,46)(20,33,31,35)(24,38,27,42)(25,39,32,40)(28,44,43,45)(37,47,41,48)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=h,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154807);