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 | |
4270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(3,7)(9,10,15,17)(11,16,21,23)(12,18,20,14)(13,19,22,24)(25,26)(27,35)(28,33)(29,38)(30,32)(31,34)(36,40)(37,39);(1,2)(3,4)(5,6)(7,8)(9,11)(10,16)(12,22)(13,20)(14,19)(15,21)(17,23)(18,24)(25,27,31,30)(26,32,34,35)(28,36,29,37)(33,39,38,40);(1,2)(3,8)(4,7)(5,6)(10,17)(11,21)(12,20)(19,24)(25,28,31,29)(26,33,34,38)(27,36,30,37)(32,39,35,40);(1,3,5,7)(2,4,6,8)(9,12,15,20)(10,18,17,14)(11,22,21,13)(16,24,23,19)(25,29,31,28)(26,33,34,38)(27,37,30,36)(32,39,35,40);(1,4,5,8)(2,7,6,3)(9,13,15,22)(10,19,17,24)(11,12,21,20)(14,16,18,23)(25,26)(27,32)(28,33)(29,38)(30,35)(31,34)(36,39)(37,40);(1,2)(3,4)(5,6)(7,8)(9,14,15,18)(10,20,17,12)(11,19,21,24)(13,23,22,16)(25,30,31,27)(26,32,34,35)(28,37,29,36)(33,39,38,40)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=ghi,d^2=gi,e^2=h,f^2=gh,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, 10050529);