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 | #44889 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,12)(13,21)(14,17)(15,19)(16,24)(18,22)(20,23)(26,32)(28,38)(33,35)(36,39);(2,3)(6,7)(9,11,14,19)(10,12,17,15)(13,18,24,23)(16,20,21,22)(25,26)(27,33)(28,34)(29,35)(30,36)(31,32)(37,38)(39,40);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,21)(23,24)(25,27,31,29)(26,33,32,35)(28,36,38,39)(30,37,40,34);(1,4)(2,6)(3,7)(5,8)(9,12,14,15)(10,11,17,19)(13,22,24,20)(16,23,21,18)(25,28)(26,34)(27,36)(29,39)(30,33)(31,38)(32,37)(35,40);(9,11)(10,12)(13,23)(14,19)(15,17)(16,22)(18,24)(20,21)(25,29,31,27)(26,35,32,33)(28,36,38,39)(30,37,40,34);(9,13,14,24)(10,16,17,21)(11,18,19,23)(12,20,15,22)(25,30)(26,36)(27,37)(28,35)(29,34)(31,40)(32,39)(33,38)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=g,d^2=h,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9138063);