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 | #35696 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4326 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)(10,15)(11,17)(16,22)(18,23);(3,4)(7,8)(9,10)(11,12)(13,16)(14,15)(17,19)(18,20)(21,22)(23,24)(25,26,30,34)(27,32,37,31)(28,29,33,40)(35,39,38,36);(1,3)(2,7)(4,6)(5,8)(11,17)(12,19)(13,21)(16,22)(25,27,30,37)(26,31,34,32)(28,38,33,35)(29,39,40,36);(3,4)(7,8)(9,11,14,17)(10,12,15,19)(13,18,21,23)(16,20,22,24)(25,28,30,33)(26,29,34,40)(27,35,37,38)(31,36,32,39);(1,4,6,3)(2,7,5,8)(9,12)(10,11)(13,20)(14,19)(15,17)(16,18)(21,24)(22,23)(25,27)(26,32)(28,38)(29,36)(30,37)(31,34)(33,35)(39,40);(1,5,6,2)(3,7,4,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,33)(27,36)(28,34)(30,40)(31,38)(32,35)(37,39)Polycyclic presentation:
2^9|b^2=hi,c^2=hi,d^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8904244);