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 | |
4478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,7,4)(9,10)(11,12)(13,17)(14,15)(16,19)(18,20)(21,24)(22,23)(25,26)(27,30)(28,35)(29,32)(31,36)(33,37)(34,40)(38,39);(1,3,5,7)(2,6,8,4)(10,15)(12,16)(17,23)(20,21)(27,33)(30,37)(34,39)(38,40);(1,4)(2,7)(3,8)(5,6)(25,26,29,32)(27,30,33,37)(28,31,36,35)(34,38,39,40);(1,3,5,7)(2,6,8,4)(9,11)(10,16)(12,15)(13,18)(14,19)(17,21)(20,23)(22,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(1,4)(2,3)(5,6)(7,8)(9,12,14,16)(10,11,15,19)(13,21,22,20)(17,24,23,18)(25,27,29,33)(26,30,32,37)(28,34,36,39)(31,38,35,40);(1,2)(3,6)(4,7)(5,8)(9,13,14,22)(10,17,15,23)(11,18,19,24)(12,20,16,21)(25,28)(26,31)(27,34)(29,36)(30,38)(32,35)(33,39)(37,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=hi,d^2=i,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9782703);