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 | |
4505 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,15,18)(11,16,20,14)(12,21,13,17)(19,24,22,23)(25,26,30,33)(27,31,37,40)(28,32,38,29)(34,35,36,39);(2,5)(6,7)(9,11)(10,16)(12,22)(13,19)(14,18)(15,20)(17,24)(21,23)(25,27)(26,31)(28,36)(29,35)(30,37)(32,39)(33,40)(34,38);(2,5)(3,8)(9,12,15,13)(10,17,18,21)(11,19,20,22)(14,24,16,23)(25,28,30,38)(26,32,33,29)(27,34,37,36)(31,35,40,39);(2,5)(6,7)(9,13,15,12)(10,17,18,21)(11,19,20,22)(14,23,16,24)(25,29)(26,28)(27,35)(30,32)(31,36)(33,38)(34,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,14)(10,11)(12,23)(13,24)(15,16)(17,22)(18,20)(19,21)(25,28)(26,32)(27,36)(29,33)(30,38)(31,39)(34,37)(35,40);(1,3,4,8)(2,6,5,7)(9,11)(10,14)(12,22)(13,19)(15,20)(16,18)(17,23)(21,24)(25,27)(26,31)(28,36)(29,35)(30,37)(32,39)(33,40)(34,38)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8851043);