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 | #36714 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,6)(7,8)(9,10,14,16)(11,17,20,15)(12,21,23,13)(18,24,22,19)(25,26,31,28)(27,32,34,30)(29,33,39,36)(35,40,37,38);(2,6)(3,7)(9,11,14,20)(10,15,16,17)(12,22,23,18)(13,19,21,24)(25,27,31,34)(26,32,28,30)(29,37,39,35)(33,38,36,40);(1,3,5,7)(2,4,6,8)(10,16)(13,21)(15,17)(19,24)(27,34)(30,32)(35,37)(38,40);(1,4)(2,7)(3,6)(5,8)(9,11)(10,15)(12,22)(13,19)(14,20)(16,17)(18,23)(21,24)(25,28,31,26)(27,30,34,32)(29,33,39,36)(35,40,37,38);(9,12)(10,13)(11,18)(14,23)(15,24)(16,21)(17,19)(20,22)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37);(9,13)(10,12)(11,19)(14,21)(15,22)(16,23)(17,18)(20,24)(25,30)(26,27)(28,34)(29,38)(31,32)(33,35)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=gh,d^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8267131);