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 | #35276 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4431 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)(3,7)(9,10,15,18)(11,16,13,23)(12,14,22,17)(19,24,21,20)(25,26)(27,32)(28,36)(29,34)(30,31)(33,38)(35,37)(39,40);(1,2,5,4)(3,6,7,8)(9,11,15,13)(10,16,18,23)(12,21,22,19)(14,20,17,24)(26,31)(27,34)(33,36)(35,40);(3,7)(6,8)(9,12,15,22)(10,17,18,14)(11,19,13,21)(16,20,23,24)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,39,36,37);(1,3)(2,6)(4,8)(5,7)(9,12)(10,14)(11,19)(13,21)(15,22)(16,24)(17,18)(20,23)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(9,13,15,11)(10,16,18,23)(12,21,22,19)(14,24,17,20);(1,4,5,2)(3,8,7,6)(9,14)(10,12)(11,20)(13,24)(15,17)(16,21)(18,22)(19,23)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,40,36,35)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=i,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014866);