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 | |
4585 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,4)(6,7)(9,10)(11,19)(12,23)(13,18)(14,15)(16,22)(17,21)(20,24)(25,26,31,34)(27,32,35,30)(28,29,33,40)(36,37,38,39);(1,2,5,8)(3,6,4,7)(9,11,15,22)(10,16,14,19)(12,13,21,20)(17,18,23,24)(25,27,31,35)(26,32,34,30)(28,38,33,36)(29,39,40,37);(1,3,5,4)(2,6,8,7)(10,14)(12,21)(13,20)(16,19)(27,35)(28,33)(29,40)(30,32);(9,12)(10,17)(11,20)(13,22)(14,23)(15,21)(16,24)(18,19)(25,28)(26,29)(27,36)(30,39)(31,33)(32,37)(34,40)(35,38);(1,2)(3,6)(4,7)(5,8)(9,13,15,20)(10,18,14,24)(11,21,22,12)(16,23,19,17)(25,29)(26,33)(27,37)(28,34)(30,36)(31,40)(32,38)(35,39);(1,4,5,3)(2,7,8,6)(9,14,15,10)(11,19,22,16)(12,23,21,17)(13,24,20,18)(25,30)(26,27)(28,39)(29,36)(31,32)(33,37)(34,35)(38,40)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8969303);