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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4453 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)(3,4)(5,6)(7,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,21)(20,23)(22,24)(25,26,30,33)(27,28,31,34)(29,32,38,40)(35,36,39,37);(1,3,6,7)(2,4,5,8)(10,15)(11,18)(17,23)(19,24)(25,27)(26,28)(29,35)(30,31)(32,36)(33,34)(37,40)(38,39);(1,4,6,8)(2,7,5,3)(12,20)(17,23)(19,24)(21,22)(27,31)(28,34)(35,39)(36,37);(1,2,6,5)(3,8,7,4)(9,11,14,18)(10,16,15,13)(12,19,20,24)(17,21,23,22)(25,26)(27,34)(28,31)(29,32)(30,33)(35,37)(36,39)(38,40);(1,4)(2,3)(5,7)(6,8)(9,12,14,20)(10,17,15,23)(11,19,18,24)(13,22,16,21)(25,28,30,34)(26,31,33,27)(29,37,38,36)(32,35,40,39);(1,5,6,2)(3,8,7,4)(9,13,14,16)(10,18,15,11)(12,21,20,22)(17,19,23,24)(25,29,30,38)(26,32,33,40)(27,35,31,39)(28,36,34,37)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,d^2=gh,e^2=ghi,f^2=gi,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, 9983134);