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 | |
4443 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)(11,19)(12,21)(13,18)(14,22)(15,17)(16,20)(23,24)(25,26,31,29)(27,32,37,39)(28,34,38,33)(30,40,36,35);(1,2)(3,4)(5,7)(6,8)(9,11,15,20)(10,16,17,19)(12,13,22,23)(14,18,21,24)(25,27,31,37)(26,32,29,39)(28,36,38,30)(33,40,34,35);(1,3)(2,6)(4,7)(5,8)(10,17)(11,20)(12,22)(18,24)(25,28,31,38)(26,33,29,34)(27,30,37,36)(32,35,39,40);(1,3)(2,4)(5,8)(6,7)(9,12)(10,14)(11,13)(15,22)(16,18)(17,21)(19,24)(20,23)(25,28)(26,34)(27,36)(29,33)(30,37)(31,38)(32,35)(39,40);(1,4,5,6)(2,3,7,8)(9,13)(10,18)(11,12)(14,16)(15,23)(17,24)(19,21)(20,22)(25,29,31,26)(27,32,37,39)(28,34,38,33)(30,35,36,40);(1,3)(2,4)(5,8)(6,7)(9,14)(10,12)(11,18)(13,16)(15,21)(17,22)(19,23)(20,24)(25,30,31,36)(26,35,29,40)(27,28,37,38)(32,33,39,34)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,e^2=gi,f^2=gh,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, 10052877);