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 | #2428 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4358 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)(11,13)(12,17)(14,19)(15,16)(18,23)(20,22)(21,24)(25,26,31,32)(27,34,28,33)(29,30,36,35)(37,38,39,40);(1,2)(3,4)(5,6)(7,8)(9,11,15,18)(10,13,16,23)(12,20,21,14)(17,22,24,19)(26,32)(29,36)(33,34)(37,39);(2,6)(4,7)(10,16)(11,18)(14,20)(17,24)(25,27,31,28)(26,33,32,34)(29,37,36,39)(30,40,35,38);(2,6)(3,8)(9,12)(10,17)(11,20)(13,22)(14,18)(15,21)(16,24)(19,23)(25,28,31,27)(26,34,32,33)(29,37,36,39)(30,40,35,38);(1,3)(2,7)(4,6)(5,8)(9,13,15,23)(10,18,16,11)(12,19,21,22)(14,17,20,24)(25,29,31,36)(26,35,32,30)(27,37,28,39)(33,38,34,40);(1,4,5,7)(2,3,6,8)(9,14,15,20)(10,19,16,22)(11,12,18,21)(13,17,23,24)(25,30,31,35)(26,36,32,29)(27,38,28,40)(33,37,34,39)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,d^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=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8306215);