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 | |
4306 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)(3,4)(5,8)(6,7)(9,10)(11,16)(12,17)(13,21)(14,15)(18,23)(19,20)(22,24)(25,26)(27,35)(28,32)(29,33)(30,34)(31,37)(36,40)(38,39);(1,2)(3,6)(4,7)(5,8)(10,15)(12,16)(18,21)(20,22)(25,27,30,37)(26,31,34,35)(28,29,36,39)(32,38,40,33);(1,2,5,8)(3,4,7,6)(10,15)(11,17)(18,21)(19,24);(1,3)(2,6)(4,8)(5,7)(9,11,14,17)(10,16,15,12)(13,19,23,24)(18,22,21,20)(25,28,30,36)(26,32,34,40)(27,29,37,39)(31,38,35,33);(1,3)(2,4)(5,7)(6,8)(9,12,14,16)(10,17,15,11)(13,22,23,20)(18,19,21,24)(25,29)(26,33)(27,36)(28,37)(30,39)(31,32)(34,38)(35,40);(1,4)(2,7)(3,8)(5,6)(9,13,14,23)(10,18,15,21)(11,19,17,24)(12,20,16,22)(25,26)(27,31)(28,32)(29,38)(30,34)(33,39)(35,37)(36,40)Polycyclic presentation:
2^9|b^2=ghi,c^2=i,d^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9636410);