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 | #57013 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,16)(11,13,12,19)(14,17,22,23)(18,21,20,24)(25,26,31,30)(27,34,32,28)(29,37,38,33)(35,36,39,40);(1,2,5,6)(3,7,8,4)(10,16)(13,19)(17,23)(21,24)(25,27,31,32)(26,28,30,34)(29,35,38,39)(33,36,37,40);(2,6)(4,7)(9,11,15,12)(10,13,16,19)(14,20,22,18)(17,24,23,21)(25,28,31,34)(26,32,30,27)(29,36,38,40)(33,39,37,35);(1,3,5,8)(2,7,6,4)(9,12,15,11)(10,13,16,19)(14,20,22,18)(17,21,23,24)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,40)(38,39);(1,3)(2,4)(5,8)(6,7)(9,13)(10,12)(11,16)(14,21)(15,19)(17,20)(18,23)(22,24)(25,29,31,38)(26,33,30,37)(27,35,32,39)(28,36,34,40);(1,4)(2,3)(5,7)(6,8)(9,14,15,22)(10,17,16,23)(11,18,12,20)(13,21,19,24)(25,30,31,26)(27,34,32,28)(29,37,38,33)(35,40,39,36)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=g,d^2=gi,e^2=h,f^2=g,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, 8265690);