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 | #2537 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4254 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(16,17)(18,21)(23,24)(25,26,32,31)(27,37,39,33)(28,40,38,34)(29,36,30,35);(1,2)(3,7)(4,8)(5,6)(9,10,14,15)(11,16,12,17)(13,18,22,21)(19,23,20,24)(25,27)(26,33)(28,29)(30,38)(31,37)(32,39)(34,36)(35,40);(1,3)(2,4)(5,8)(6,7)(9,11,14,12)(10,16,15,17)(13,19,22,20)(18,23,21,24)(25,28,32,38)(26,34,31,40)(27,29,39,30)(33,36,37,35);(2,6)(4,7)(9,10)(11,16)(12,17)(13,21)(14,15)(18,22)(19,24)(20,23)(25,29,32,30)(26,35,31,36)(27,28,39,38)(33,40,37,34);(1,3)(2,7)(4,6)(5,8)(9,12,14,11)(10,17,15,16)(13,19,22,20)(18,23,21,24)(25,30,32,29)(26,36,31,35)(27,38,39,28)(33,34,37,40);(1,4,5,7)(2,8,6,3)(9,13,14,22)(10,18,15,21)(11,19,12,20)(16,23,17,24)(25,31,32,26)(27,37,39,33)(28,40,38,34)(29,35,30,36)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8606350);