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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4647 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)(3,7)(4,8)(5,6)(9,10,13,17)(11,16,20,14)(12,21,22,15)(18,23,19,24)(25,26,31,34)(27,30,32,28)(29,33,38,40)(35,39,37,36);(2,6)(3,8)(9,11)(10,14)(12,18)(13,20)(15,23)(16,17)(19,22)(21,24)(25,27)(26,30)(28,34)(29,37)(31,32)(33,36)(35,38)(39,40);(1,3)(2,4)(5,8)(6,7)(9,12)(10,15)(11,18)(13,22)(14,23)(16,24)(17,21)(19,20)(25,28,31,30)(26,32,34,27)(29,36,38,39)(33,37,40,35);(1,3)(2,4)(5,8)(6,7)(9,12)(10,15)(11,19)(13,22)(14,24)(16,23)(17,21)(18,20)(25,29)(26,33)(27,35)(28,36)(30,39)(31,38)(32,37)(34,40);(1,4)(2,3)(5,7)(6,8)(9,10)(11,14)(12,15)(13,17)(16,20)(18,23)(19,24)(21,22)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,38)(39,40);(9,11)(10,16)(12,18)(13,20)(14,17)(15,24)(19,22)(21,23)(25,30,31,28)(26,32,34,27)(29,36,38,39)(33,35,40,37)Polycyclic presentation:
2^9|a^2=gi,c^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8570110);