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 | |
4561 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,19)(12,18)(13,16)(14,20)(15,17)(21,22)(23,24)(25,26)(27,32)(28,38)(29,31)(30,34)(33,39)(35,40)(36,37);(1,3,5,8)(2,6,4,7)(9,11,15,13)(10,16,17,19)(12,21,20,23)(14,22,18,24)(25,27)(26,32)(28,30)(29,36)(31,37)(33,40)(34,38)(35,39);(10,17)(12,20)(16,19)(21,23)(25,28,31,39)(26,33,29,38)(27,35,37,30)(32,34,36,40);(1,2,5,4)(3,6,8,7)(9,12,15,20)(10,18,17,14)(11,21,13,23)(16,24,19,22)(25,28)(26,33)(27,30)(29,38)(31,39)(32,40)(34,36)(35,37);(9,13,15,11)(10,19,17,16)(12,23,20,21)(14,24,18,22)(25,29,31,26)(27,36,37,32)(28,38,39,33)(30,34,35,40);(1,4,5,2)(3,7,8,6)(9,14,15,18)(10,20,17,12)(11,22,13,24)(16,23,19,21)(25,30)(26,34)(27,28)(29,40)(31,35)(32,38)(33,36)(37,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=i,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7808023);