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 | |
4441 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,7,4)(9,10,15,17)(11,16,14,24)(12,18,21,13)(19,20,22,23)(25,26,31,35)(27,36,34,30)(28,39,38,32)(29,33,37,40);(1,3,5,7)(2,6,8,4)(9,11,15,14)(10,16,17,24)(12,19,21,22)(13,23,18,20)(25,27,31,34)(26,30,35,36)(28,37,38,29)(32,40,39,33);(1,3)(2,4)(5,7)(6,8)(10,17)(12,21)(16,24)(19,22)(25,28,31,38)(26,32,35,39)(27,37,34,29)(30,40,36,33);(1,3,5,7)(2,6,8,4)(9,12)(10,18)(11,19)(13,17)(14,22)(15,21)(16,20)(23,24)(25,28)(26,32)(27,37)(29,34)(30,40)(31,38)(33,36)(35,39);(1,3)(2,6)(4,8)(5,7)(9,13,15,18)(10,12,17,21)(11,20,14,23)(16,22,24,19)(25,29,31,37)(26,33,35,40)(27,38,34,28)(30,39,36,32);(1,4,5,6)(2,7,8,3)(9,14,15,11)(10,16,17,24)(12,22,21,19)(13,23,18,20)(25,30)(26,34)(27,35)(28,40)(29,32)(31,36)(33,38)(37,39)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=i,e^2=hi,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, 10075576);