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 | |
4557 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)(3,4)(5,6)(7,8)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,26)(27,30)(28,33)(29,32)(31,36)(34,37)(35,39)(38,40);(2,6)(3,7)(9,11)(10,14)(12,20)(13,19)(15,17)(16,24)(18,23)(21,22)(26,27)(31,33)(32,34)(38,39);(1,3,5,7)(2,4,6,8)(10,13)(11,17)(16,21)(18,20)(25,26,30,27)(28,31,36,33)(29,32,37,34)(35,38,40,39);(1,4)(2,7)(3,6)(5,8)(9,12,15,23)(10,16,13,21)(11,18,17,20)(14,22,19,24)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(9,13,15,10)(11,19,17,14)(12,21,23,16)(18,24,20,22)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(9,14,15,19)(10,17,13,11)(12,22,23,24)(16,20,21,18)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)Polycyclic presentation:
2^9|c^2=g,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8027110);