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 | |
4494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,8)(4,7)(5,6)(26,31)(27,35)(33,40)(34,37);(1,2,6,5)(3,7,4,8)(9,10,14,12)(11,15,18,19)(13,16,21,20)(17,22,23,24)(25,26,30,31)(27,32,35,39)(28,33,36,40)(29,34,38,37);(1,3)(2,7)(4,6)(5,8)(10,12)(13,21)(15,19)(17,23)(28,36)(29,38)(33,40)(34,37);(1,2,6,5)(3,7,4,8)(9,11,14,18)(10,15,12,19)(13,17,21,23)(16,22,20,24)(25,27)(26,32)(28,37)(29,33)(30,35)(31,39)(34,36)(38,40);(1,4,6,3)(2,7,5,8)(9,12,14,10)(11,15,18,19)(13,20,21,16)(17,22,23,24)(25,28,30,36)(26,33,31,40)(27,34,35,37)(29,32,38,39);(1,5,6,2)(3,7,4,8)(9,13)(10,16)(11,17)(12,20)(14,21)(15,22)(18,23)(19,24)(25,29,30,38)(26,34,31,37)(27,33,35,40)(28,32,36,39)Polycyclic presentation:
2^9|b^2=gi,d^2=hi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8870904);