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 | #53610 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,6,8,7)(9,10)(11,16)(12,15)(13,17)(14,22)(18,23)(19,20)(21,24)(25,26)(27,30)(28,35)(29,32)(31,36)(33,37)(34,40)(38,39);(1,2)(3,7)(4,5)(6,8)(9,11,15,19)(10,16,12,20)(13,23,24,14)(17,18,21,22)(27,33)(30,37)(34,39)(38,40);(1,2)(3,7)(4,5)(6,8)(10,12)(11,19)(13,24)(18,22)(25,26,29,32)(27,30,33,37)(28,31,36,35)(34,38,39,40);(1,3)(2,6)(4,7)(5,8)(9,12,15,10)(11,16,19,20)(13,21,24,17)(14,22,23,18)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(1,4,5,2)(3,7,8,6)(9,13,15,24)(10,17,12,21)(11,14,19,23)(16,22,20,18)(25,27)(26,30)(28,34)(29,33)(31,38)(32,37)(35,40)(36,39);(1,3)(2,7)(4,6)(5,8)(9,14)(10,18)(11,13)(12,22)(15,23)(16,21)(17,20)(19,24)(25,28,29,36)(26,31,32,35)(27,34,33,39)(30,38,37,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=ghi,d^2=i,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9993536);