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 |
| prime radical | #44103 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4245 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,5,7)(9,10,15,17)(11,16,12,22)(13,18,19,14)(20,21,23,24)(25,26)(27,35)(28,30)(29,33)(31,32)(34,40)(36,37)(38,39);(1,3,6,5)(2,7,4,8)(9,11)(10,16)(12,15)(13,23)(14,21)(17,22)(18,24)(19,20)(25,27,31,33)(26,29,32,35)(28,38,34,36)(30,37,40,39);(1,3,6,5)(2,8,4,7)(10,17)(11,12)(13,19)(21,24)(26,32)(28,34)(29,35)(36,38);(1,2,6,4)(3,8,5,7)(9,12,15,11)(10,16,17,22)(13,23,19,20)(14,24,18,21)(25,28,31,34)(26,30,32,40)(27,36,33,38)(29,39,35,37);(1,4,6,2)(3,8,5,7)(9,13)(10,18)(11,20)(12,23)(14,17)(15,19)(16,21)(22,24)(25,29,31,35)(26,33,32,27)(28,39,34,37)(30,38,40,36);(1,5,6,3)(2,7,4,8)(9,14,15,18)(10,19,17,13)(11,21,12,24)(16,20,22,23)(25,30)(26,34)(27,37)(28,32)(29,36)(31,40)(33,39)(35,38)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=ghi,d^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9954661);