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 | |
4413 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,15,16)(11,17,12,14)(13,21,23,18)(19,22,20,24)(27,35)(28,37)(31,38)(32,40);(1,2,5,4)(3,6,8,7)(10,16)(11,12)(18,21)(19,20)(25,26,30,34)(27,31,35,38)(28,32,37,40)(29,33,36,39);(2,4)(6,7)(9,11,15,12)(10,14,16,17)(13,19,23,20)(18,22,21,24)(25,27)(26,31)(28,36)(29,37)(30,35)(32,39)(33,40)(34,38);(9,12,15,11)(10,17,16,14)(13,20,23,19)(18,24,21,22)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,3,5,8)(2,6,4,7)(9,13,15,23)(10,18,16,21)(11,19,12,20)(14,22,17,24)(25,28,30,37)(26,32,34,40)(27,36,35,29)(31,39,38,33);(1,4,5,2)(3,7,8,6)(9,14,15,17)(10,11,16,12)(13,22,23,24)(18,19,21,20)(25,29,30,36)(26,33,34,39)(27,37,35,28)(31,40,38,32)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=g,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8087618);