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 | |
4360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(9,10,15,17)(11,21,22,16)(12,13,19,18)(14,24,23,20)(25,26,30,33)(27,29,35,31)(28,32,36,39)(34,37,40,38);(1,2,4,5)(3,6,7,8)(9,11)(10,16)(12,23)(13,24)(14,19)(15,22)(17,21)(18,20)(28,36)(32,39)(34,40)(37,38);(1,2)(3,6)(4,5)(7,8)(10,17)(11,22)(12,19)(20,24)(25,27,30,35)(26,31,33,29)(28,34,36,40)(32,38,39,37);(2,5)(6,8)(9,12,15,19)(10,18,17,13)(11,23,22,14)(16,20,21,24)(25,27)(26,29)(28,34)(30,35)(31,33)(32,37)(36,40)(38,39);(1,2)(3,8)(4,5)(6,7)(9,13,15,18)(10,19,17,12)(11,20,22,24)(14,16,23,21)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(1,3)(2,6)(4,7)(5,8)(9,14,15,23)(10,20,17,24)(11,19,22,12)(13,16,18,21)(25,29)(26,27)(28,37)(30,31)(32,34)(33,35)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10219825);