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 | |
4403 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,17)(12,14)(13,16)(15,18)(19,24)(20,22)(21,23);(1,3)(2,4)(5,7)(6,8)(10,12)(11,18)(16,20)(19,21)(25,26,30,33)(27,34,37,31)(28,29,32,38)(35,40,39,36);(2,6)(3,7)(10,12)(15,17)(16,20)(23,24)(25,27,30,37)(26,31,33,34)(28,35,32,39)(29,36,38,40);(3,7)(4,8)(9,11)(10,15)(12,17)(13,21)(14,18)(16,24)(19,22)(20,23)(25,28)(26,29)(27,35)(30,32)(31,36)(33,38)(34,40)(37,39);(1,2)(3,8)(4,7)(5,6)(9,12,14,10)(11,17,18,15)(13,20,22,16)(19,24,21,23)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,34)(38,40);(1,4,5,8)(2,7,6,3)(9,13,14,22)(10,16,12,20)(11,19,18,21)(15,23,17,24)(25,29)(26,32)(27,36)(28,33)(30,38)(31,39)(34,35)(37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8514690);