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 | |
4514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,16,20,24)(12,21,13,14)(18,19,22,23)(25,26)(27,28)(29,37)(30,34)(31,32)(33,40)(35,38)(36,39);(1,2)(3,4)(5,7)(6,8)(9,11,15,20)(10,16,17,24)(12,22,13,18)(14,19,21,23)(25,27)(26,31)(28,34)(29,35)(30,32)(33,38)(36,37)(39,40);(1,3)(2,6)(4,7)(5,8)(9,12,15,13)(10,14,17,21)(11,18,20,22)(16,23,24,19)(25,28)(26,32)(27,34)(29,36)(30,31)(33,39)(35,37)(38,40);(1,2)(3,6)(4,8)(5,7)(9,13,15,12)(10,14,17,21)(11,18,20,22)(16,19,24,23)(25,29,30,40)(26,33,34,37)(27,35,32,39)(28,36,31,38);(1,4,5,6)(2,3,7,8)(9,11)(10,16)(12,18)(13,22)(14,23)(15,20)(17,24)(19,21)(25,28)(26,27)(29,38)(30,31)(32,34)(33,39)(35,37)(36,40);(1,3)(2,4)(5,8)(6,7)(9,14)(10,13)(11,19)(12,17)(15,21)(16,18)(20,23)(22,24)(25,27)(26,31)(28,34)(29,39)(30,32)(33,36)(35,40)(37,38)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,d^2=h,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,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, 9136652);