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 | #2733 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4129 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)(3,4,8,6)(9,10,15,12)(11,16,14,18)(13,17,22,20)(19,23,21,24)(25,26)(27,36)(28,37)(29,31)(30,34)(32,35)(33,38)(39,40);(1,3,5,8)(2,6,7,4)(10,12)(16,18)(17,20)(23,24)(25,27)(26,31)(28,34)(29,32)(30,33)(35,36)(37,40)(38,39);(1,3)(2,4)(5,8)(6,7)(9,11,15,14)(10,16,12,18)(13,21,22,19)(17,24,20,23)(25,28,32,38)(26,33,35,37)(27,34,29,39)(30,36,40,31);(1,4)(2,3)(5,6)(7,8)(9,12,15,10)(11,18,14,16)(13,20,22,17)(19,24,21,23)(25,29,32,27)(26,31,35,36)(28,34,38,39)(30,33,40,37);(9,13,15,22)(10,17,12,20)(11,19,14,21)(16,23,18,24)(25,30,32,40)(26,34,35,39)(27,37,29,33)(28,31,38,36);(9,14,15,11)(10,18,12,16)(13,19,22,21)(17,23,20,24)(25,31)(26,27)(28,30)(29,35)(32,36)(33,34)(37,39)(38,40)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=gi,d^2=gi,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, 9993521);