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 | #43113 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4543 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)(3,7)(4,5)(6,8)(9,10)(11,14)(12,19)(13,16)(15,20)(17,21)(18,24)(22,23)(25,26,30,33)(27,29,35,31)(28,32,38,40)(34,39,36,37);(1,2,5,4)(3,6,8,7)(11,17)(14,21)(18,23)(22,24)(25,27)(26,29)(28,36)(30,35)(31,33)(32,37)(34,38)(39,40);(1,3)(2,6)(4,7)(5,8)(9,10,13,16)(11,14,17,21)(12,15,20,19)(18,22,23,24)(25,27,30,35)(26,31,33,29)(28,34,38,36)(32,37,40,39);(1,3,5,8)(2,6,4,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,27)(26,29)(28,36)(30,35)(31,33)(32,37)(34,38)(39,40);(1,3)(2,7)(4,6)(5,8)(9,11,13,17)(10,14,16,21)(12,18,20,23)(15,22,19,24)(25,28)(26,32)(27,34)(29,39)(30,38)(31,37)(33,40)(35,36);(1,4,5,2)(3,7,8,6)(9,12)(10,15)(11,18)(13,20)(14,22)(16,19)(17,23)(21,24)(25,29)(26,27)(28,37)(30,31)(32,36)(33,35)(34,40)(38,39)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gh,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10055005);