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 | |
4498 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)(12,15)(13,16)(17,18)(19,24)(20,23)(21,22)(26,32)(27,30)(28,34)(37,39);(2,6)(3,7)(9,11)(10,14)(12,18)(13,21)(15,17)(16,22)(19,23)(20,24)(25,26)(27,35)(28,29)(30,33)(31,32)(34,40)(36,37)(38,39);(1,3,5,7)(2,4,6,8)(10,12)(11,17)(16,20)(19,21)(25,27,31,30)(26,33,32,35)(28,38,34,36)(29,37,40,39);(1,4)(2,7)(3,6)(5,8)(9,12,15,10)(11,18,17,14)(13,20,23,16)(19,22,21,24)(25,28,31,34)(26,29,32,40)(27,36,30,38)(33,39,35,37);(9,13,15,23)(10,16,12,20)(11,19,17,21)(14,24,18,22)(25,29)(26,34)(27,37)(28,32)(30,39)(31,40)(33,36)(35,38);(9,14,15,18)(10,17,12,11)(13,22,23,24)(16,19,20,21)(25,30,31,27)(26,33,32,35)(28,38,34,36)(29,39,40,37)Polycyclic presentation:
2^9|c^2=g,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9624786);