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 | |
4632 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)(3,4)(5,6)(7,8)(10,14)(15,17)(16,21)(22,23);(2,6)(4,7)(9,10,13,14)(11,17,18,15)(12,16,20,21)(19,23,24,22)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,39)(38,40);(1,2,5,6)(3,7,8,4)(11,18)(12,20)(15,17)(16,21)(26,30)(28,37)(31,33)(34,38);(1,3)(2,7)(4,6)(5,8)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,27)(26,31)(28,38)(29,35)(30,33)(32,40)(34,37)(36,39);(1,4,5,7)(2,3,6,8)(9,11,13,18)(10,15,14,17)(12,19,20,24)(16,22,21,23)(25,28)(26,32)(27,34)(29,37)(30,36)(31,39)(33,40)(35,38);(1,3)(2,7)(4,6)(5,8)(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,37)(38,40)Polycyclic presentation:
2^9|b^2=gh,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 8947542);