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 | |
4709 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,12)(11,15,19,17)(13,20,21,16)(18,23,24,22)(26,30)(28,36)(31,33)(34,40);(1,2,5,8)(3,4,7,6)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,3)(2,6)(4,8)(5,7)(10,12)(15,17)(16,20)(22,23)(25,26,29,30)(27,31,35,33)(28,32,36,37)(34,38,40,39);(1,4)(2,7)(3,8)(5,6)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,3)(2,4)(5,7)(6,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(1,4)(2,7)(3,8)(5,6)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=g,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10004353);