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 | #37839 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4375 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,15,18)(11,16,12,13)(14,17,22,24)(19,23,20,21)(25,26,31,33)(27,29,28,35)(30,38,39,32)(34,40,36,37);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,13,18,16)(14,20,22,19)(17,23,24,21)(25,27,31,28)(26,29,33,35)(30,34,39,36)(32,37,38,40);(11,12)(13,16)(19,20)(21,23)(25,28,31,27)(26,29,33,35)(30,36,39,34)(32,37,38,40);(1,2,5,4)(3,6,8,7)(9,12,15,11)(10,16,18,13)(14,20,22,19)(17,23,24,21)(25,29)(26,28)(27,33)(30,37)(31,35)(32,36)(34,38)(39,40);(1,4,5,2)(3,7,8,6)(9,13,15,16)(10,11,18,12)(14,21,22,23)(17,19,24,20)(25,30)(26,32)(27,34)(28,36)(29,37)(31,39)(33,38)(35,40);(9,14,15,22)(10,17,18,24)(11,19,12,20)(13,21,16,23)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7650828);