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 | |
4341 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,17)(12,20)(13,16)(14,19)(15,21)(18,22)(23,24)(25,26,31,28)(27,34,30,32)(29,33,37,36)(35,40,38,39);(1,3,5,8)(2,6,7,4)(9,10,13,16)(11,14,19,17)(12,15,21,20)(18,22,24,23)(29,37)(33,36)(35,38)(39,40);(1,2)(3,4)(5,7)(6,8)(9,10,13,16)(11,14,19,17)(12,20,21,15)(18,23,24,22)(25,27,31,30)(26,32,28,34)(29,35,37,38)(33,39,36,40);(1,4)(2,3)(5,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,28,31,26)(27,34,30,32)(29,36,37,33)(35,40,38,39);(1,3)(2,4)(5,8)(6,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,21)(22,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,37)(32,39)(34,40);(1,4)(2,3)(5,6)(7,8)(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24)(25,30,31,27)(26,34,28,32)(29,38,37,35)(33,40,36,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=hi,d^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8851700);