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 | |
4513 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,15,17)(11,16,12,22)(13,18,23,21)(14,20,24,19)(25,26,31,33)(27,35,37,32)(28,30,39,40)(29,34,36,38);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,15)(13,20)(14,21)(17,22)(18,24)(19,23)(25,27,31,37)(26,32,33,35)(28,38,39,34)(29,30,36,40);(10,17)(11,12)(18,21)(19,20)(26,33)(28,39)(32,35)(34,38);(1,2,5,4)(3,7,8,6)(9,12,15,11)(10,16,17,22)(13,19,23,20)(14,18,24,21)(25,28)(26,30)(27,34)(29,32)(31,39)(33,40)(35,36)(37,38);(1,3)(2,7)(4,6)(5,8)(9,13)(10,18)(11,20)(12,19)(14,22)(15,23)(16,24)(17,21)(25,29)(26,34)(27,30)(28,32)(31,36)(33,38)(35,39)(37,40);(1,4,5,2)(3,7,8,6)(9,14,15,24)(10,19,17,20)(11,21,12,18)(13,16,23,22)(25,30)(26,28)(27,36)(29,37)(31,40)(32,38)(33,39)(34,35)Polycyclic presentation:
2^9|a^2=gi,b^2=g,d^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9845271);