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 |
| prime radical | #51896 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,6,8,4)(9,10)(11,16)(12,20)(13,18)(14,15)(17,21)(19,23)(22,24)(25,26,31,33)(27,32,28,37)(29,40,34,30)(35,39,38,36);(1,3,5,8)(2,6,7,4)(10,15)(11,18)(12,21)(23,24)(25,27)(26,32)(28,31)(29,35)(30,36)(33,37)(34,38)(39,40);(1,3)(2,6)(4,7)(5,8)(11,18)(13,16)(19,22)(23,24)(26,33)(27,28)(30,40)(35,38);(2,7)(3,8)(9,11,14,18)(10,16,15,13)(12,22,21,19)(17,23,20,24)(25,28,31,27)(26,32,33,37)(29,35,34,38)(30,39,40,36);(1,3)(2,4)(5,8)(6,7)(9,12)(10,17)(11,19)(13,23)(14,21)(15,20)(16,24)(18,22)(25,29,31,34)(26,30,33,40)(27,35,28,38)(32,36,37,39);(1,4)(2,3)(5,6)(7,8)(9,13,14,16)(10,18,15,11)(12,23,21,24)(17,22,20,19)(25,30)(26,34)(27,36)(28,39)(29,33)(31,40)(32,38)(35,37)Polycyclic presentation:
2^9|a^2=i,b^2=hi,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8841764);