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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4321 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,6,8,4)(9,10)(11,15)(12,13)(14,16)(17,18)(19,20)(21,24)(22,23)(25,26,32,34)(27,30,31,28)(29,33,39,40)(35,38,37,36);(1,3,5,8)(2,6,7,4)(9,11)(10,15)(12,23)(13,22)(14,20)(16,19)(17,24)(18,21)(25,27,32,31)(26,28,34,30)(29,37,39,35)(33,38,40,36);(1,3)(2,6)(4,7)(5,8)(10,16)(11,20)(12,18)(22,24)(25,28,32,30)(26,27,34,31)(29,38,39,36)(33,37,40,35);(2,7)(3,8)(9,12)(10,17)(11,21)(13,16)(14,18)(15,22)(19,24)(20,23)(25,29,32,39)(26,33,34,40)(27,35,31,37)(28,36,30,38);(1,3)(2,4)(5,8)(6,7)(9,13,14,17)(10,18,16,12)(11,22,20,24)(15,21,19,23)(25,30,32,28)(26,27,34,31)(29,36,39,38)(33,37,40,35);(1,4)(2,3)(5,6)(7,8)(9,11)(10,19)(12,21)(13,22)(14,20)(15,16)(17,24)(18,23)(25,31,32,27)(26,28,34,30)(29,37,39,35)(33,36,40,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=g,d^2=g,e^2=gh,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, 8841691);