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 | #45616 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4272 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)(3,5)(4,7)(6,8)(13,21)(16,20)(18,24)(22,23)(25,26,31,36)(27,29,34,32)(28,35,30,33)(37,40,38,39);(1,3,4,8)(2,5,7,6)(9,10,14,12)(11,17,19,15)(13,16,21,20)(18,23,24,22)(25,27)(26,32)(28,37)(29,36)(30,38)(31,34)(33,40)(35,39);(1,3)(2,6)(4,8)(5,7)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27)(26,29)(28,37)(30,38)(31,34)(32,36)(33,39)(35,40);(1,2)(3,6)(4,7)(5,8)(25,28,31,30)(26,33,36,35)(27,37,34,38)(29,39,32,40);(9,12,14,10)(11,15,19,17)(13,20,21,16)(18,22,24,23)(25,29)(26,34)(27,36)(28,39)(30,40)(31,32)(33,38)(35,37);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,28)(26,35,36,33)(27,38,34,37)(29,40,32,39)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=hi,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8002521);