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 | |
4357 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,16,20,14)(12,18,22,23)(13,19,21,24)(25,26)(27,36)(28,38)(29,35)(30,32)(31,33)(34,39)(37,40);(1,2)(3,8)(4,6)(5,7)(9,11)(10,16)(12,21)(13,22)(14,17)(15,20)(18,24)(19,23)(25,27,31,30)(26,32,33,36)(28,37,39,35)(29,34,40,38);(1,3,5,4)(2,6,7,8)(10,17)(11,20)(12,22)(19,24)(26,33)(28,39)(32,36)(35,37);(1,2)(3,6)(4,8)(5,7)(9,12,15,22)(10,18,17,23)(11,21,20,13)(14,19,16,24)(25,28)(26,34)(27,37)(29,36)(30,35)(31,39)(32,40)(33,38);(1,4,5,3)(2,6,7,8)(9,13,15,21)(10,19,17,24)(11,12,20,22)(14,23,16,18)(25,29,31,40)(26,35,33,37)(27,38,30,34)(28,32,39,36);(1,2)(3,6)(4,8)(5,7)(9,14)(10,20)(11,17)(12,19)(13,18)(15,16)(21,23)(22,24)(25,30,31,27)(26,36,33,32)(28,35,39,37)(29,38,40,34)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=ghi,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10063384);