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 | #39856 | 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) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,21)(12,13)(14,18)(15,19)(16,20)(17,24)(22,23)(25,26,32,30)(27,33,29,28)(31,34,39,38)(35,40,37,36);(1,2)(3,6)(4,7)(5,8)(9,11)(10,17)(12,19)(13,23)(14,22)(15,18)(16,24)(20,21)(25,27,32,29)(26,28,30,33)(31,35,39,37)(34,36,38,40);(1,3,5,4)(2,6,8,7)(9,12,16,14)(10,13,20,18)(11,22,24,19)(15,21,23,17)(25,28)(26,29)(27,30)(31,36)(32,33)(34,37)(35,38)(39,40);(1,3,5,4)(2,6,8,7)(9,13,16,18)(10,12,20,14)(11,23,24,15)(17,19,21,22)(25,29,32,27)(26,33,30,28)(31,37,39,35)(34,40,38,36);(1,3,5,4)(2,6,8,7)(9,14,16,12)(10,18,20,13)(11,22,24,19)(15,21,23,17)(25,30,32,26)(27,33,29,28)(31,34,39,38)(35,36,37,40);(1,4,5,3)(2,7,8,6)(9,15,16,23)(10,19,20,22)(11,18,24,13)(12,21,14,17)(25,31,32,39)(26,34,30,38)(27,35,29,37)(28,36,33,40)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7561408);