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 | |
4313 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,20)(12,16)(13,17)(14,18)(15,19)(21,22)(23,24)(25,26)(27,36)(28,38)(29,34)(30,35)(31,32)(33,39)(37,40);(2,3)(6,7)(9,11,15,18)(10,14,19,20)(12,21,17,24)(13,22,16,23)(25,27)(26,29)(28,37)(30,38)(31,36)(32,34)(33,40)(35,39);(1,3,5,2)(4,7,8,6)(9,12,15,17)(10,16,19,13)(11,21,18,24)(14,23,20,22)(25,28,32,39)(26,33,31,38)(27,37,34,35)(29,40,36,30);(1,4)(2,6)(3,7)(5,8)(9,12)(10,16)(11,21)(13,19)(14,23)(15,17)(18,24)(20,22)(25,29,32,36)(26,34,31,27)(28,30,39,40)(33,37,38,35);(9,13,15,16)(10,17,19,12)(11,22,18,23)(14,24,20,21)(25,30)(26,35)(27,33)(28,29)(31,37)(32,40)(34,38)(36,39);(9,14)(10,18)(11,19)(12,23)(13,21)(15,20)(16,24)(17,22)(25,31,32,26)(27,36,34,29)(28,38,39,33)(30,35,40,37)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=g,d^2=h,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9113782);