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 | |
4543 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)(9,10,14,15)(11,16,18,13)(12,20,21,17)(19,23,22,24)(25,26)(27,30)(28,33)(29,34)(31,36)(32,37)(35,38)(39,40);(1,3,5,8)(2,6,7,4)(10,15)(11,18)(12,21)(23,24)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,38)(39,40);(1,3)(2,6)(4,7)(5,8)(11,18)(13,16)(19,22)(23,24)(26,27)(29,37)(31,33)(35,40);(2,7)(3,8)(9,11)(10,16)(12,22)(13,15)(14,18)(17,23)(19,21)(20,24)(25,27,30,26)(28,31,36,33)(29,34,37,32)(35,38,40,39);(1,3)(2,4)(5,8)(6,7)(9,12,14,21)(10,17,15,20)(11,19,18,22)(13,23,16,24)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(1,4)(2,3)(5,6)(7,8)(9,13,14,16)(10,11,15,18)(12,23,21,24)(17,19,20,22)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,d^2=h,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, 8841786);