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 | |
4730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(11,17)(16,23)(18,20)(25,26)(27,32)(28,33)(29,38)(30,31)(34,40)(35,39)(36,37);(1,2,5,4)(3,7,8,6)(9,10,14,15)(11,13,17,22)(12,16,19,23)(18,21,20,24)(25,26)(27,32)(28,36)(29,34)(30,31)(33,37)(35,39)(38,40);(1,3,5,8)(2,6,4,7)(11,17)(12,19)(13,22)(16,23)(26,31)(27,35)(28,37)(29,40);(3,8)(6,7)(9,11,14,17)(10,13,15,22)(12,20,19,18)(16,24,23,21)(25,27,30,35)(26,32,31,39)(28,38,37,34)(29,36,40,33);(1,3)(2,7)(4,6)(5,8)(9,12)(10,16)(11,18)(13,21)(14,19)(15,23)(17,20)(22,24)(25,28)(26,33)(27,34)(29,39)(30,37)(31,36)(32,40)(35,38);(1,4,5,2)(3,6,8,7)(9,13)(10,17)(11,15)(12,21)(14,22)(16,20)(18,23)(19,24)(25,29,30,40)(26,34,31,38)(27,36,35,33)(28,39,37,32)Polycyclic presentation:
2^9|b^2=ghi,c^2=hi,d^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8935526);