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 | |
4545 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,18)(12,20)(13,15)(14,17)(16,21)(19,23)(22,24)(25,26,29,31)(27,33,35,30)(28,32,38,37)(34,40,36,39);(2,6)(3,8)(9,11,14,13)(10,15,17,18)(12,19,21,22)(16,23,20,24)(25,27)(26,30)(28,36)(29,35)(31,33)(32,40)(34,38)(37,39);(1,3)(2,4)(5,8)(6,7)(12,21)(16,20)(19,22)(23,24)(26,31)(30,33)(32,37)(39,40);(1,4)(2,3)(5,7)(6,8)(9,12)(10,16)(11,19)(13,22)(14,21)(15,23)(17,20)(18,24)(25,26)(27,30)(28,37)(29,31)(32,38)(33,35)(34,40)(36,39);(1,2)(3,4)(5,6)(7,8)(9,13,14,11)(10,15,17,18)(12,19,21,22)(16,24,20,23)(25,28)(26,32)(27,34)(29,38)(30,39)(31,37)(33,40)(35,36);(1,4)(2,3)(5,7)(6,8)(9,10)(11,15)(12,16)(13,18)(14,17)(19,23)(20,21)(22,24)(25,27)(26,30)(28,36)(29,35)(31,33)(32,40)(34,38)(37,39)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9034753);