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 | |
4550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,17)(11,12,19,15)(13,16,23,24)(18,20,22,21)(25,26,30,31)(27,35,37,32)(28,36,39,33)(29,34,40,38);(1,2)(3,4)(5,7)(6,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,21)(20,24)(22,23)(26,31)(29,40)(32,35)(33,36);(1,3)(2,6)(4,7)(5,8)(25,27)(26,32)(28,34)(29,36)(30,37)(31,35)(33,40)(38,39);(1,4,5,6)(2,3,7,8)(9,12,14,15)(10,11,17,19)(13,21,23,20)(16,22,24,18)(25,28,30,39)(26,33,31,36)(27,34,37,38)(29,32,40,35);(1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,22)(14,19)(15,17)(16,21)(18,23)(20,24)(25,29,30,40)(26,34,31,38)(27,36,37,33)(28,35,39,32);(1,2)(3,6)(4,8)(5,7)(9,13,14,23)(10,16,17,24)(11,18,19,22)(12,20,15,21)(25,27)(26,32)(28,38)(29,33)(30,37)(31,35)(34,39)(36,40)Polycyclic presentation:
2^9|a^2=g,d^2=gi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9131394);