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 | #2430 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,16)(12,20)(13,17)(14,19)(15,21)(18,22)(23,24)(25,26,31,33)(27,36,30,32)(28,29,35,34)(37,38,39,40);(1,2)(3,8)(4,6)(5,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,31,30)(26,32,33,36)(28,37,35,39)(29,40,34,38);(1,3,5,6)(2,4,7,8)(26,33)(28,35)(32,36)(37,39);(9,12,13,21)(10,15,17,20)(11,18,19,24)(14,22,16,23)(25,28,31,35)(26,34,33,29)(27,37,30,39)(32,38,36,40);(1,2)(3,4)(5,7)(6,8)(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,22)(21,24)(25,29,31,34)(26,35,33,28)(27,38,30,40)(32,37,36,39);(1,4)(2,6)(3,7)(5,8)(9,10)(11,14)(12,15)(13,17)(16,19)(18,22)(20,21)(23,24)(25,30,31,27)(26,36,33,32)(28,39,35,37)(29,38,34,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=i,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9591178);