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 | |
4500 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)(5,8)(6,7)(9,10,15,12)(11,19,20,16)(13,17,24,21)(14,22,23,18)(28,36)(32,39)(35,37)(38,40);(1,3,5,7)(2,4,8,6)(9,11,15,20)(10,16,12,19)(13,23,24,14)(17,22,21,18)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,39)(37,38);(1,3)(2,6)(4,8)(5,7)(10,12)(16,19)(17,21)(18,22)(25,27,30,29)(26,31,33,34)(28,35,36,37)(32,38,39,40);(1,2,5,8)(3,4,7,6)(9,12,15,10)(11,19,20,16)(13,21,24,17)(14,22,23,18)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,39)(34,40);(1,3)(2,4)(5,7)(6,8)(9,13,15,24)(10,17,12,21)(11,14,20,23)(16,18,19,22)(25,26)(27,31)(28,32)(29,34)(30,33)(35,38)(36,39)(37,40);(1,4)(2,7)(3,8)(5,6)(9,14)(10,18)(11,13)(12,22)(15,23)(16,17)(19,21)(20,24)(25,29,30,27)(26,31,33,34)(28,37,36,35)(32,38,39,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=ghi,d^2=h,e^2=g,f^2=ghi,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, 9633363);