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 | |
4447 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)(3,4)(5,7)(6,8)(9,10,15,17)(11,16,20,14)(12,18,22,23)(13,19,21,24)(26,29)(27,33)(32,36)(34,37);(1,3,6,5)(2,7,8,4)(9,11,15,20)(10,16,17,14)(12,21,22,13)(18,24,23,19)(25,26,30,29)(27,31,33,35)(28,36,38,32)(34,40,37,39);(10,17)(11,20)(12,22)(19,24)(25,27,30,33)(26,31,29,35)(28,34,38,37)(32,39,36,40);(1,4,6,7)(2,5,8,3)(9,12)(10,18)(11,21)(13,20)(14,19)(15,22)(16,24)(17,23)(25,27)(26,31)(28,37)(29,35)(30,33)(32,40)(34,38)(36,39);(1,5,6,3)(2,7,8,4)(9,13,15,21)(10,19,17,24)(11,22,20,12)(14,18,16,23)(25,28)(26,32)(27,34)(29,36)(30,38)(31,39)(33,37)(35,40);(1,2)(3,7)(4,5)(6,8)(9,14,15,16)(10,20,17,11)(12,19,22,24)(13,23,21,18)(25,29,30,26)(27,35,33,31)(28,32,38,36)(34,39,37,40)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=hi,d^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10050800);