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 | |
4216 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)(11,15)(12,16)(13,24)(14,18)(17,21)(19,20)(22,23)(25,26,31,33)(27,34,37,30)(28,32,38,29)(35,36,40,39);(1,2,5,4)(3,7,8,6)(9,11,14,20)(10,15,18,19)(12,21,23,13)(16,17,22,24)(25,27)(26,30)(28,35)(29,36)(31,37)(32,39)(33,34)(38,40);(1,3,5,8)(2,6,4,7)(9,10,14,18)(11,19,20,15)(12,22,23,16)(13,24,21,17)(25,28,31,38)(26,32,33,29)(27,35,37,40)(30,39,34,36);(1,3,5,8)(2,7,4,6)(9,10)(11,15)(12,22)(13,17)(14,18)(16,23)(19,20)(21,24)(25,26)(27,30)(28,32)(29,38)(31,33)(34,37)(35,39)(36,40);(1,3)(2,7)(4,6)(5,8)(9,12)(10,16)(11,13)(14,23)(15,24)(17,19)(18,22)(20,21)(25,29,31,32)(26,28,33,38)(27,36,37,39)(30,35,34,40);(1,4,5,2)(3,6,8,7)(9,13)(10,17)(11,12)(14,21)(15,22)(16,19)(18,24)(20,23)(25,30)(26,27)(28,36)(29,35)(31,34)(32,40)(33,37)(38,39)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gi,d^2=hi,e^2=g,f^2=hi,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, 9514859);