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 | |
4433 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,16)(11,15,19,22)(12,17,21,24)(13,20,23,18)(25,26,30,34)(27,32,37,31)(28,29,33,40)(35,36,38,39);(1,2)(3,5)(4,7)(6,8)(9,11,14,19)(10,15,16,22)(12,20,21,18)(13,17,23,24)(25,27,30,37)(26,31,34,32)(28,38,33,35)(29,36,40,39);(1,3)(2,7)(4,6)(5,8)(10,16)(11,19)(12,21)(13,23)(25,27,30,37)(26,32,34,31)(28,38,33,35)(29,39,40,36);(2,8)(3,4)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,21)(23,24)(25,27)(26,31)(28,38)(29,36)(30,37)(32,34)(33,35)(39,40);(1,4,6,3)(2,5,8,7)(9,12)(10,17)(11,20)(13,22)(14,21)(15,23)(16,24)(18,19)(25,28,30,33)(26,29,34,40)(27,35,37,38)(31,39,32,36);(1,5)(2,3)(4,8)(6,7)(9,13,14,23)(10,18,16,20)(11,17,19,24)(12,22,21,15)(25,29)(26,33)(27,36)(28,34)(30,40)(31,35)(32,38)(37,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8970441);