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 |
| prime radical | #47870 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,7)(3,8,4,5)(9,10)(11,18)(12,15)(13,20)(14,17)(16,22)(19,21)(23,24)(25,26,31,35)(27,28,38,32)(29,33,30,34)(36,39,37,40);(2,7)(3,4)(10,12)(11,17)(16,20)(19,23)(25,27,31,38)(26,32,35,28)(29,36,30,37)(33,40,34,39);(2,7)(5,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23)(25,27)(26,28)(29,36)(30,37)(31,38)(32,35)(33,39)(34,40);(1,3,6,4)(2,5,7,8)(9,12,15,10)(11,18,17,14)(13,20,22,16)(19,24,23,21)(25,28)(26,27)(29,39)(30,40)(31,32)(33,36)(34,37)(35,38);(1,4,6,3)(2,8,7,5)(9,13,15,22)(10,16,12,20)(11,19,17,23)(14,21,18,24)(25,29,31,30)(26,33,35,34)(27,36,38,37)(28,39,32,40);(1,5,6,8)(2,3,7,4)(9,14,15,18)(10,17,12,11)(13,21,22,24)(16,23,20,19)(25,30,31,29)(26,34,35,33)(27,37,38,36)(28,40,32,39)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8109475);