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 | #55680 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,18)(11,19,17,13)(12,22,14,16)(20,24,21,23)(25,26,31,35)(27,29,36,32)(28,33,37,38)(30,34,39,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,17)(10,13,18,19)(12,20,14,21)(16,24,22,23)(27,36)(28,37)(29,32)(33,38);(2,3)(6,7)(9,12,15,14)(10,16,18,22)(11,20,17,21)(13,24,19,23)(25,27,31,36)(26,32,35,29)(28,30,37,39)(33,40,38,34);(4,8)(6,7)(9,10)(11,19)(12,16)(13,17)(14,22)(15,18)(20,23)(21,24)(25,28)(26,33)(27,30)(29,34)(31,37)(32,40)(35,38)(36,39);(1,3,5,2)(4,7,8,6)(9,13,15,19)(10,17,18,11)(12,23,14,24)(16,20,22,21)(25,29,31,32)(26,27,35,36)(28,34,37,40)(30,38,39,33);(1,4)(2,6)(3,7)(5,8)(9,14,15,12)(10,16,18,22)(11,21,17,20)(13,24,19,23)(25,30)(26,34)(27,37)(28,36)(29,38)(31,39)(32,33)(35,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8519367);