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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4036 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,20,14,16)(12,13,19,18)(21,22,23,24)(25,26,31,36)(27,30,35,32)(28,33,38,40)(29,39,37,34);(1,2)(3,8)(4,6)(5,7)(9,11,15,14)(10,16,17,20)(12,21,19,23)(13,24,18,22)(25,27)(26,32)(28,37)(29,38)(30,36)(31,35)(33,39)(34,40);(1,3,5,6)(2,4,7,8)(10,17)(12,19)(16,20)(21,23)(25,28,31,38)(26,33,36,40)(27,37,35,29)(30,34,32,39);(9,12,15,19)(10,18,17,13)(11,21,14,23)(16,22,20,24)(25,29,31,37)(26,34,36,39)(27,38,35,28)(30,33,32,40);(1,2)(3,4)(5,7)(6,8)(9,13,15,18)(10,19,17,12)(11,22,14,24)(16,21,20,23)(25,27)(26,30)(28,37)(29,38)(31,35)(32,36)(33,34)(39,40);(1,4)(2,6)(3,7)(5,8)(9,14,15,11)(10,20,17,16)(12,23,19,21)(13,22,18,24)(25,30)(26,35)(27,36)(28,34)(29,33)(31,32)(37,40)(38,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=ghi,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, 9591336);