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 | |
4385 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,4,8,7)(9,10,15,17)(11,16,13,14)(12,18,22,24)(19,21,20,23)(25,26,31,30)(27,32,34,29)(28,33,37,38)(35,39,40,36);(1,3)(2,7)(4,5)(6,8)(9,11)(10,16)(12,20)(13,15)(14,17)(18,23)(19,22)(21,24)(25,27)(26,32)(28,35)(29,30)(31,34)(33,39)(36,38)(37,40);(1,4,6,7)(2,3,5,8)(10,17)(11,13)(18,24)(19,20)(27,34)(29,32)(35,40)(36,39);(2,5)(3,8)(9,12)(10,18)(11,19)(13,20)(14,23)(15,22)(16,21)(17,24)(25,28)(26,33)(27,35)(29,36)(30,38)(31,37)(32,39)(34,40);(1,4)(2,3)(5,8)(6,7)(9,13,15,11)(10,16,17,14)(12,20,22,19)(18,21,24,23)(25,29)(26,27)(28,36)(30,34)(31,32)(33,35)(37,39)(38,40);(1,5,6,2)(3,4,8,7)(9,14,15,16)(10,11,17,13)(12,21,22,23)(18,20,24,19)(25,30,31,26)(27,29,34,32)(28,33,37,38)(35,39,40,36)Polycyclic presentation:
2^9|a^2=gi,c^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8541700);