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 | |
4723 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(4,8)(9,10,15,13)(11,12,19,16)(14,21,22,17)(18,23,24,20)(25,26,29,32)(27,33,35,30)(28,31,37,40)(34,39,36,38);(2,6)(3,7)(9,11,15,19)(10,16,13,12)(14,18,22,24)(17,23,21,20)(25,27)(26,30)(28,36)(29,35)(31,39)(32,33)(34,37)(38,40);(1,2)(3,8)(4,7)(5,6)(10,13)(12,16)(14,22)(18,24)(25,27,29,35)(26,30,32,33)(28,34,37,36)(31,38,40,39);(2,6)(3,7)(9,12)(10,11)(13,19)(14,20)(15,16)(17,18)(21,24)(22,23)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,37)(39,40);(1,3)(2,4)(5,7)(6,8)(9,13,15,10)(11,16,19,12)(14,21,22,17)(18,23,24,20)(25,28,29,37)(26,31,32,40)(27,34,35,36)(30,38,33,39);(1,4)(2,3)(5,8)(6,7)(9,14)(10,17)(11,18)(12,20)(13,21)(15,22)(16,23)(19,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,39)(37,40)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9401963);