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 | #45041 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4453 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,17)(12,14)(13,16)(15,19)(18,24)(20,22)(21,23)(25,26,32,29)(27,33,31,35)(28,30,39,34)(36,40,38,37);(1,2)(3,6)(4,7)(5,8)(10,12)(15,17)(16,20)(23,24)(25,27,32,31)(26,33,29,35)(28,38,39,36)(30,37,34,40);(1,3)(2,4)(5,7)(6,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,23)(20,24)(21,22)(25,28,32,39)(26,34,29,30)(27,36,31,38)(33,37,35,40);(1,3)(2,6)(4,8)(5,7)(9,12,14,10)(11,17,19,15)(13,16,22,20)(18,23,21,24)(25,29,32,26)(27,33,31,35)(28,30,39,34)(36,37,38,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,17)(13,21)(14,19)(16,24)(18,22)(20,23)(25,30)(26,28)(27,37)(29,39)(31,40)(32,34)(33,38)(35,36);(1,4,5,6)(2,7,8,3)(9,13,14,22)(10,16,12,20)(11,18,19,21)(15,23,17,24)(25,31,32,27)(26,35,29,33)(28,38,39,36)(30,37,34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8584314);