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 | |
4472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(10,15)(11,16)(17,22)(18,23)(25,26)(27,36)(28,30)(29,39)(31,35)(32,38)(33,37)(34,40);(1,3,5,8)(2,6,7,4)(9,10,14,15)(11,12,16,20)(13,17,21,22)(18,19,23,24)(25,27,31,38)(26,32,35,36)(28,29,37,40)(30,34,33,39);(1,3)(2,4)(5,8)(6,7)(13,21)(17,22)(18,23)(19,24)(25,28,31,37)(26,33,35,30)(27,29,38,40)(32,39,36,34);(1,4)(2,3)(5,6)(7,8)(9,11)(10,12)(13,18)(14,16)(15,20)(17,19)(21,23)(22,24)(25,27)(26,32)(28,29)(30,34)(31,38)(33,39)(35,36)(37,40);(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,29)(26,34)(27,37)(28,38)(30,36)(31,40)(32,33)(35,39);(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,19,20,24)(25,30,31,33)(26,28,35,37)(27,34,38,39)(29,36,40,32)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8230491);