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 | #34589 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,5,7)(9,10,14,15)(11,17,12,16)(13,21,22,18)(19,23,20,24)(25,26,31,32)(27,28,35,40)(29,33,30,34)(36,38,37,39);(1,3,6,5)(2,7,4,8)(10,15)(16,17)(18,21)(23,24)(26,32)(27,35)(33,34)(36,37);(1,2,6,4)(3,7,5,8)(9,11,14,12)(10,16,15,17)(13,19,22,20)(18,23,21,24)(25,27)(26,28)(29,36)(30,37)(31,35)(32,40)(33,38)(34,39);(2,4)(7,8)(9,10)(11,17)(12,16)(13,18)(14,15)(19,24)(20,23)(21,22)(25,28)(26,27)(29,38)(30,39)(31,40)(32,35)(33,36)(34,37);(1,4,6,2)(3,8,5,7)(9,12,14,11)(10,17,15,16)(13,20,22,19)(18,24,21,23)(25,29,31,30)(26,33,32,34)(27,36,35,37)(28,38,40,39);(1,5,6,3)(2,8,4,7)(9,13)(10,18)(11,19)(12,20)(14,22)(15,21)(16,23)(17,24)(25,30,31,29)(26,34,32,33)(27,37,35,36)(28,39,40,38)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8065994);