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 | |
4477 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)(18,23)(21,34)(24,28)(25,36)(30,43)(33,35)(38,40)(45,46);(1,3,5,7)(2,4,6,8)(10,13)(14,15)(18,23)(19,27)(20,31)(26,37)(30,43)(33,35)(38,40)(47,48);(9,10)(11,14)(12,13)(15,16)(17,18,22,23)(19,28,27,24)(20,25,31,36)(21,26,34,37)(29,40,32,38)(30,44,43,39)(33,41,35,42)(45,48,46,47);(2,6)(3,7)(9,11,12,16)(10,14,13,15)(17,19,22,27)(18,24,23,28)(20,32,31,29)(21,30,34,43)(25,40,36,38)(26,39,37,44)(33,46,35,45)(41,48,42,47);(1,2)(3,4)(5,6)(7,8)(17,20,22,31)(18,25,23,36)(19,29,27,32)(21,35,34,33)(24,38,28,40)(26,42,37,41)(30,46,43,45)(39,48,44,47);(1,4)(2,7)(3,6)(5,8)(17,21)(18,26)(19,30)(20,33)(22,34)(23,37)(24,39)(25,41)(27,43)(28,44)(29,45)(31,35)(32,46)(36,42)(38,47)(40,48)Polycyclic presentation:
2^9|b^2=ghi,c^2=h,d^2=i,e^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, 10154190);