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 | |
4509 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,11)(15,17)(16,18)(23,24)(25,26,30,32)(27,34,37,31)(28,36,38,33)(29,39,40,35);(1,2,5,8)(3,4,7,6)(9,10,14,11)(12,15,20,17)(13,16,22,18)(19,23,21,24)(25,27,30,37)(26,31,32,34)(28,35,38,39)(29,36,40,33);(1,3)(2,6)(4,8)(5,7)(26,32)(28,38)(31,34)(35,39);(1,4)(2,7)(3,8)(5,6)(9,11,14,10)(12,15,20,17)(13,18,22,16)(19,23,21,24)(25,28,30,38)(26,33,32,36)(27,35,37,39)(29,34,40,31);(1,3)(2,4)(5,7)(6,8)(9,12)(10,15)(11,17)(13,21)(14,20)(16,24)(18,23)(19,22)(25,27)(26,34)(28,39)(29,33)(30,37)(31,32)(35,38)(36,40);(1,4)(2,7)(3,8)(5,6)(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,29)(26,35)(27,36)(28,34)(30,40)(31,38)(32,39)(33,37)Polycyclic presentation:
2^9|a^2=h,b^2=gi,d^2=g,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, 10003724);