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 | #45801 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4189 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)(3,7)(4,5)(6,8)(9,10,14,17)(11,15,18,22)(12,13,16,21)(19,20,23,24)(25,26,31,29)(27,35,36,32)(28,34,30,33)(37,39,38,40);(3,8)(6,7)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(26,29)(27,36)(33,34)(37,38);(1,3,5,8)(2,6,4,7)(11,18)(12,16)(13,21)(15,22)(25,27,31,36)(26,32,29,35)(28,37,30,38)(33,39,34,40);(3,8)(6,7)(9,12)(10,13)(11,19)(14,16)(15,20)(17,21)(18,23)(22,24)(25,28,31,30)(26,33,29,34)(27,37,36,38)(32,39,35,40);(1,3)(2,7)(4,6)(5,8)(9,13)(10,16)(11,20)(12,17)(14,21)(15,23)(18,24)(19,22)(25,29,31,26)(27,35,36,32)(28,34,30,33)(37,40,38,39);(1,4,5,2)(3,6,8,7)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,30,31,28)(26,34,29,33)(27,38,36,37)(32,40,35,39)Polycyclic presentation:
2^9|a^2=ghi,c^2=h,d^2=g,e^2=g,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, 8881757);