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 | |
4500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,18)(11,13,17,15)(12,16,22,20)(19,23,24,21)(25,26)(27,34)(28,32)(29,37)(30,31)(33,39)(35,36)(38,40);(1,2,5,8)(3,4,7,6)(9,11,14,17)(10,15,18,13)(12,19,22,24)(16,21,20,23)(26,31)(28,34)(33,37)(36,38);(1,3)(2,6)(4,8)(5,7)(25,27,30,32)(26,28,31,34)(29,35,39,40)(33,36,37,38);(1,4)(2,7)(3,8)(5,6)(9,10)(11,15)(12,20)(13,17)(14,18)(16,22)(19,23)(21,24)(25,27)(26,28)(29,35)(30,32)(31,34)(33,36)(37,38)(39,40);(1,3)(2,4)(5,7)(6,8)(9,12)(10,16)(11,19)(13,23)(14,22)(15,21)(17,24)(18,20)(25,28)(26,32)(27,31)(29,38)(30,34)(33,35)(36,39)(37,40);(1,4)(2,7)(3,8)(5,6)(9,13,14,15)(10,17,18,11)(12,21,22,23)(16,19,20,24)(25,29,30,39)(26,33,31,37)(27,35,32,40)(28,36,34,38)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=h,f^2=g,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, 10002112);