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 | |
4388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(25,26,31,32)(27,29,28,35)(30,33,38,39)(34,37,36,40);(3,7)(6,8)(9,10,14,15)(11,16,19,23)(12,20,22,13)(17,18,21,24)(26,32)(29,35)(33,39)(37,40);(1,2,5,4)(3,6,7,8)(10,15)(13,20)(16,23)(18,24)(25,27,31,28)(26,29,32,35)(30,36,38,34)(33,40,39,37);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,21)(13,18)(14,19)(15,23)(17,22)(20,24)(25,28,31,27)(26,29,32,35)(30,36,38,34)(33,37,39,40);(1,4,5,2)(3,8,7,6)(9,12)(10,13)(11,17)(14,22)(15,20)(16,24)(18,23)(19,21)(25,29)(26,28)(27,32)(30,37)(31,35)(33,36)(34,39)(38,40);(9,13)(10,12)(11,18)(14,20)(15,22)(16,21)(17,23)(19,24)(25,30)(26,33)(27,34)(28,36)(29,37)(31,38)(32,39)(35,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gh,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8258836);