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 | #37332 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4440 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)(11,16,13,14)(12,18,22,24)(19,21,20,23)(25,26,30,34)(27,35,37,31)(28,32,38,40)(29,33,36,39);(1,2,4,5)(3,6,7,8)(9,11)(10,16)(12,20)(13,15)(14,17)(18,23)(19,22)(21,24)(25,27)(26,31)(28,29)(30,37)(32,39)(33,40)(34,35)(36,38);(2,5)(6,8)(10,17)(11,13)(18,24)(19,20)(25,28,30,38)(26,32,34,40)(27,29,37,36)(31,39,35,33);(3,7)(6,8)(9,12)(10,18)(11,19)(13,20)(14,23)(15,22)(16,21)(17,24)(25,26)(27,31)(28,32)(29,39)(30,34)(33,36)(35,37)(38,40);(1,2)(3,6)(4,5)(7,8)(9,13,15,11)(10,16,17,14)(12,20,22,19)(18,21,24,23)(25,28)(26,32)(27,36)(29,37)(30,38)(31,33)(34,40)(35,39);(1,3,4,7)(2,6,5,8)(9,14,15,16)(10,11,17,13)(12,21,22,23)(18,20,24,19)(25,29,30,36)(26,33,34,39)(27,28,37,38)(31,40,35,32)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8543224);