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 | |
4538 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)(11,20)(12,13)(14,24)(15,17)(16,21)(18,23)(19,22)(25,26,31,34)(27,32,38,29)(28,30,40,33)(35,39,36,37);(2,3)(6,7)(9,11,15,21)(10,16,17,20)(12,22,23,14)(13,24,18,19)(25,27,31,38)(26,32,34,29)(28,36,40,35)(30,37,33,39);(1,2,5,3)(4,6,8,7)(10,17)(11,21)(12,23)(19,24)(25,28,31,40)(26,33,34,30)(27,35,38,36)(29,39,32,37);(9,12)(10,18)(11,22)(13,17)(14,21)(15,23)(16,19)(20,24)(25,28)(26,30)(27,36)(29,39)(31,40)(32,37)(33,34)(35,38);(1,3,5,2)(4,6,8,7)(9,13,15,18)(10,12,17,23)(11,19,21,24)(14,20,22,16)(25,29)(26,27)(28,39)(30,36)(31,32)(33,35)(34,38)(37,40);(1,4)(2,6)(3,7)(5,8)(9,14,15,22)(10,19,17,24)(11,12,21,23)(13,20,18,16)(25,30)(26,28)(27,37)(29,35)(31,33)(32,36)(34,40)(38,39)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9755748);