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 | #42547 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4445 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,16)(11,17,12,15)(13,18,21,23)(19,24,20,22)(25,26,31,33)(27,36,39,32)(28,34,40,30)(29,37,38,35);(1,3,5,8)(2,6,4,7)(9,11,14,12)(10,15,16,17)(13,20,21,19)(18,24,23,22)(25,27)(26,32)(28,37)(29,34)(30,38)(31,39)(33,36)(35,40);(1,2,5,4)(3,6,8,7)(10,16)(11,12)(18,23)(19,20)(26,33)(28,40)(32,36)(35,37);(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23)(25,28)(26,34)(27,37)(29,32)(30,33)(31,40)(35,39)(36,38);(1,2,5,4)(3,6,8,7)(9,12,14,11)(10,17,16,15)(13,19,21,20)(18,22,23,24)(25,29,31,38)(26,35,33,37)(27,34,39,30)(28,32,40,36);(1,4,5,2)(3,7,8,6)(9,13,14,21)(10,18,16,23)(11,19,12,20)(15,22,17,24)(25,30,31,34)(26,28,33,40)(27,38,39,29)(32,37,36,35)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7876881);