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 | |
4258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,8)(5,6)(25,26)(27,35)(28,29)(30,32)(31,34)(33,40)(36,37)(38,39);(1,2,4,7)(3,6,8,5)(9,10,14,15)(11,13,17,22)(12,16,19,23)(18,21,20,24)(25,27,31,30)(26,32,34,35)(28,38,33,36)(29,37,40,39);(1,3,4,8)(2,5,7,6)(10,15)(12,19)(13,22)(18,20)(25,28,31,33)(26,29,34,40)(27,36,30,38)(32,39,35,37);(1,2,4,7)(3,6,8,5)(9,11,14,17)(10,13,15,22)(12,20,19,18)(16,24,23,21)(25,29)(26,33)(27,37)(28,34)(30,39)(31,40)(32,36)(35,38);(1,2)(3,6)(4,7)(5,8)(9,12)(10,16)(11,18)(13,21)(14,19)(15,23)(17,20)(22,24)(25,30,31,27)(26,32,34,35)(28,38,33,36)(29,39,40,37);(1,3)(2,6)(4,8)(5,7)(9,13)(10,17)(11,15)(12,21)(14,22)(16,20)(18,23)(19,24)Polycyclic presentation:
2^9|b^2=i,c^2=gh,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9414130);