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 | |
4568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,4)(9,10,15,18)(11,12,13,16)(14,17,23,24)(19,20,21,22)(25,26,31,35)(27,32,36,38)(28,33,37,40)(29,30,34,39);(1,2,6,5)(3,8,4,7)(9,11,15,13)(10,16,18,12)(14,19,23,21)(17,22,24,20)(27,36)(28,37)(32,38)(33,40);(1,3,6,4)(2,7,5,8)(25,27)(26,32)(28,29)(30,33)(31,36)(34,37)(35,38)(39,40);(1,4,6,3)(2,8,5,7)(9,12,15,16)(10,11,18,13)(14,22,23,20)(17,21,24,19)(25,28)(26,33)(27,29)(30,32)(31,37)(34,36)(35,40)(38,39);(1,5,6,2)(3,8,4,7)(9,13,15,11)(10,16,18,12)(14,19,23,21)(17,20,24,22)(25,29)(26,30)(27,28)(31,34)(32,33)(35,39)(36,37)(38,40);(9,14)(10,17)(11,19)(12,20)(13,21)(15,23)(16,22)(18,24)(25,30)(26,34)(27,33)(28,38)(29,35)(31,39)(32,37)(36,40)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8598199);