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 | #42589 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,17,19,12)(13,16,21,22)(18,23,24,20)(25,26)(27,36)(28,32)(29,38)(30,34)(31,35)(33,39)(37,40);(1,2,5,4)(3,7,8,6)(10,15)(12,17)(16,22)(20,23)(27,28)(29,39)(32,36)(33,38);(2,4)(6,7)(13,21)(16,22)(18,24)(20,23)(25,27,31,28)(26,32,35,36)(29,37,39,34)(30,33,40,38);(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,28,31,27)(26,32,35,36)(29,34,39,37)(30,33,40,38);(1,3,5,8)(2,6,4,7)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,29,31,39)(26,33,35,38)(27,37,28,34)(30,32,40,36);(1,4,5,2)(3,7,8,6)(9,13,14,21)(10,16,15,22)(11,18,19,24)(12,20,17,23)(25,30,31,40)(26,34,35,37)(27,38,28,33)(29,32,39,36)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=h,d^2=h,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8161506);