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 | #59755 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4396 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(11,18)(13,21)(15,16)(17,20)(25,26,32,31)(27,28,34,30)(29,33,39,38)(35,36,40,37);(1,2)(3,6)(4,7)(5,8)(9,10,14,12)(11,16,18,15)(13,17,21,20)(19,23,24,22)(25,27)(26,30)(28,31)(29,35)(32,34)(33,37)(36,38)(39,40);(9,11,14,18)(10,15,12,16)(13,19,21,24)(17,22,20,23)(25,28,32,30)(26,27,31,34)(29,37,39,36)(33,40,38,35);(9,12,14,10)(11,15,18,16)(13,20,21,17)(19,22,24,23)(25,29,32,39)(26,33,31,38)(27,35,34,40)(28,36,30,37);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,15)(13,19)(14,18)(17,23)(20,22)(21,24)(25,30,32,28)(26,34,31,27)(29,37,39,36)(33,40,38,35);(1,4,5,3)(2,7,8,6)(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,31,32,26)(27,28,34,30)(29,38,39,33)(35,36,40,37)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7616521);