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 | |
4476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,7)(3,8,5,4)(9,10)(11,16)(12,18)(13,19)(14,22)(15,17)(20,23)(21,24)(25,26)(27,37)(28,34)(29,30)(31,35)(32,36)(33,40)(38,39);(1,3,6,5)(2,4,7,8)(9,11,15,20)(10,16,17,23)(12,21,22,13)(14,19,18,24)(25,27,32,40)(26,33,36,37)(28,38,29,31)(30,39,34,35);(1,4)(2,5)(3,7)(6,8)(10,17)(11,20)(12,22)(19,24)(25,28,32,29)(26,34,36,30)(27,38,40,31)(33,35,37,39);(2,7)(4,8)(9,12)(10,18)(11,21)(13,20)(14,17)(15,22)(16,24)(19,23)(25,29,32,28)(26,34,36,30)(27,31,40,38)(33,35,37,39);(1,4)(2,3)(5,7)(6,8)(9,13,15,21)(10,19,17,24)(11,22,20,12)(14,23,18,16)(25,30,32,34)(26,29,36,28)(27,39,40,35)(31,37,38,33);(1,5,6,3)(2,8,7,4)(9,14,15,18)(10,12,17,22)(11,19,20,24)(13,16,21,23)(25,31)(26,35)(27,28)(29,40)(30,33)(32,38)(34,37)(36,39)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=ghi,d^2=ghi,e^2=gh,f^2=h,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, 10050852);