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) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,16)(11,13,17,21)(12,14,20,23)(18,19,22,24)(25,26,30,31)(27,35,38,32)(28,33,34,29)(36,37,40,39);(1,2)(3,6)(4,7)(5,8)(9,11,15,17)(10,13,16,21)(12,18,20,22)(14,19,23,24)(26,31)(28,34)(32,35)(36,40);(1,3)(2,5)(4,8)(6,7)(10,16)(11,17)(12,20)(19,24)(25,27,30,38)(26,32,31,35)(28,36,34,40)(29,37,33,39);(9,12)(10,14)(11,18)(13,19)(15,20)(16,23)(17,22)(21,24)(25,27)(26,32)(28,36)(29,37)(30,38)(31,35)(33,39)(34,40);(1,3)(2,6)(4,8)(5,7)(9,13,15,21)(10,17,16,11)(12,19,20,24)(14,22,23,18)(25,28)(26,33)(27,36)(29,31)(30,34)(32,39)(35,37)(38,40);(1,2)(3,5)(4,7)(6,8)(9,14)(10,12)(11,19)(13,18)(15,23)(16,20)(17,24)(21,22)(25,29,30,33)(26,34,31,28)(27,37,38,39)(32,40,35,36)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,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, 9576564);