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 | |
4266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,19)(11,17,20,15)(12,16,18,13)(21,22,23,24)(26,31)(27,29)(33,36)(35,37);(1,2,5,4)(3,7,8,6)(11,20)(15,17)(21,23)(22,24)(25,26,30,31)(27,32,29,34)(28,36,38,33)(35,40,37,39);(1,3,5,8)(2,6,4,7)(9,11,14,20)(10,15,19,17)(12,23,18,21)(13,24,16,22)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(3,8)(6,7)(9,12,14,18)(10,16,19,13)(11,21,20,23)(15,24,17,22)(25,27,30,29)(26,32,31,34)(28,35,38,37)(33,39,36,40);(1,3)(2,7)(4,6)(5,8)(9,11)(10,17)(12,23)(13,22)(14,20)(15,19)(16,24)(18,21)(25,28)(26,33)(27,35)(29,37)(30,38)(31,36)(32,39)(34,40);(1,4,5,2)(3,6,8,7)(9,13,14,16)(10,18,19,12)(11,22,20,24)(15,23,17,21)(25,29,30,27)(26,34,31,32)(28,37,38,35)(33,40,36,39)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8935331);