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 | #55299 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,16)(11,13)(12,20)(19,24)(25,26,31,32)(27,29,34,30)(28,36,39,33)(35,38,40,37);(1,2,5,4)(3,7,8,6)(9,10,15,16)(11,17,13,23)(12,21,20,14)(18,19,22,24)(26,32)(27,34)(33,36)(35,40);(2,4)(6,7)(9,11,15,13)(10,17,16,23)(12,18,20,22)(14,24,21,19)(25,27)(26,29)(28,35)(30,32)(31,34)(33,37)(36,38)(39,40);(9,12)(10,14)(11,18)(13,22)(15,20)(16,21)(17,24)(19,23)(25,28,31,39)(26,33,32,36)(27,35,34,40)(29,37,30,38);(1,3,5,8)(2,6,4,7)(9,13,15,11)(10,17,16,23)(12,22,20,18)(14,24,21,19)(25,29,31,30)(26,34,32,27)(28,37,39,38)(33,40,36,35);(1,4,5,2)(3,7,8,6)(9,14)(10,12)(11,19)(13,24)(15,21)(16,20)(17,22)(18,23)(25,30,31,29)(26,27,32,34)(28,38,39,37)(33,35,36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=h,d^2=gh,e^2=hi,f^2=i,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, 8161880);