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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4337 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)(3,6)(4,5)(7,8)(25,26)(27,34)(28,29)(30,31)(32,37)(33,40)(35,36)(38,39);(1,3,5,8)(2,6,4,7)(9,10,14,15)(11,16,20,23)(12,13,17,22)(18,24,21,19)(26,31)(28,33)(32,34)(35,38);(1,3)(2,7)(4,6)(5,8)(10,15)(13,22)(16,23)(19,24)(25,27,30,37)(26,32,31,34)(28,35,33,38)(29,39,40,36);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,21)(13,19)(14,20)(15,23)(17,18)(22,24)(25,27)(26,32)(28,38)(29,36)(30,37)(31,34)(33,35)(39,40);(1,3)(2,6)(4,7)(5,8)(9,12,14,17)(10,13,15,22)(11,18,20,21)(16,24,23,19)(25,28,30,33)(26,29,31,40)(27,35,37,38)(32,39,34,36);(1,4,5,2)(3,7,8,6)(9,13)(10,17)(11,19)(12,15)(14,22)(16,18)(20,24)(21,23)(25,29)(26,33)(27,36)(28,31)(30,40)(32,35)(34,38)(37,39)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=gh,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9360241);