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 | #46661 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4233 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,14,12)(11,17,19,15)(13,20,21,16)(18,22,24,23)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(1,3,5,8)(2,6,7,4)(10,12)(15,17)(16,20)(22,23)(28,36)(31,38)(34,40)(37,39);(1,2)(3,4)(5,7)(6,8)(10,12)(13,21)(15,17)(18,24)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,4)(2,3)(5,6)(7,8)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23);(1,3)(2,4)(5,8)(6,7)(9,12,14,10)(11,15,19,17)(13,20,21,16)(18,22,24,23)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40);(1,4)(2,3)(5,6)(7,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=gh,d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8870818);