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 | #42541 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4225 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(10,14)(12,17)(16,20)(18,22)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(1,2)(3,6)(4,5)(7,8)(9,10,15,14)(11,16,21,20)(12,13,17,23)(18,19,22,24)(28,36)(31,38)(34,40)(37,39);(2,5)(6,8)(9,11,15,21)(10,16,14,20)(12,22,17,18)(13,24,23,19)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,2)(3,6)(4,5)(7,8)(9,12)(10,13)(11,18)(14,23)(15,17)(16,19)(20,24)(21,22)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39);(1,2)(3,6)(4,5)(7,8)(9,13,15,23)(10,17,14,12)(11,19,21,24)(16,22,20,18)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(1,3,4,7)(2,6,5,8)(9,14,15,10)(11,20,21,16)(12,23,17,13)(18,24,22,19)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,d^2=h,e^2=g,f^2=gi,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^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8039662);