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 | #2424 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4215 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,15,16)(11,13,18,23)(12,17,21,24)(14,22,20,19)(25,26,32,36)(27,35,31,30)(28,37,39,33)(29,38,40,34);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,20)(14,21)(15,18)(16,23)(17,19)(22,24)(25,27,32,31)(26,30,36,35)(28,34,39,38)(29,37,40,33);(10,16)(11,18)(12,21)(19,22)(25,28,32,39)(26,33,36,37)(27,34,31,38)(29,35,40,30);(2,4)(6,7)(9,12,15,21)(10,17,16,24)(11,20,18,14)(13,19,23,22)(25,29,32,40)(26,34,36,38)(27,37,31,33)(28,30,39,35);(1,4,5,2)(3,7,8,6)(9,13)(10,18)(11,16)(12,22)(14,24)(15,23)(17,20)(19,21)(25,30)(26,27)(28,29)(31,36)(32,35)(33,34)(37,38)(39,40);(1,3)(2,7)(4,6)(5,8)(9,14,15,20)(10,19,16,22)(11,21,18,12)(13,17,23,24)(25,31,32,27)(26,35,36,30)(28,38,39,34)(29,33,40,37)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=hi,d^2=h,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10073463);