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 | #49604 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4386 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)(3,4)(5,6)(7,8)(9,10,15,19)(11,16,13,17)(12,14,18,24)(20,21,22,23)(25,26)(27,34)(28,36)(29,30)(31,32)(33,39)(35,38)(37,40);(2,6)(3,7)(9,11)(10,16)(12,20)(13,15)(14,21)(17,19)(18,22)(23,24)(25,27,31,30)(26,29,32,34)(28,37,39,35)(33,40,36,38);(1,3,5,7)(2,4,6,8)(11,13)(16,17)(20,22)(21,23)(26,32)(29,34)(33,36)(38,40);(1,4)(2,7)(3,6)(5,8)(9,12)(10,14)(11,20)(13,22)(15,18)(16,21)(17,23)(19,24)(25,28,31,39)(26,33,32,36)(27,35,30,37)(29,38,34,40);(9,13,15,11)(10,17,19,16)(12,22,18,20)(14,23,24,21)(25,29)(26,27)(28,38)(30,32)(31,34)(33,35)(36,37)(39,40);(9,14)(10,18)(11,21)(12,19)(13,23)(15,24)(16,22)(17,20)(25,30,31,27)(26,29,32,34)(28,35,39,37)(33,40,36,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=gh,d^2=h,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8524980);