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 | #39455 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4364 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,19)(12,17)(13,15)(14,24)(16,20)(18,21)(22,23)(26,31)(29,37)(32,38)(33,39);(1,2)(3,7)(4,8)(5,6)(9,11)(10,16)(12,21)(13,19)(14,22)(15,20)(17,24)(18,23)(25,26,30,31)(27,32,35,38)(28,33,36,39)(29,34,37,40);(1,3,5,4)(2,7,6,8)(10,13)(11,20)(12,22)(18,24)(26,31)(28,36)(32,38)(34,40);(1,3,5,4)(2,7,6,8)(9,12,15,22)(10,17,13,23)(11,21,20,14)(16,24,19,18)(25,27,30,35)(26,32,31,38)(28,34,36,40)(29,33,37,39);(9,13,15,10)(11,19,20,16)(12,23,22,17)(14,24,21,18)(25,28,30,36)(26,33,31,39)(27,34,35,40)(29,32,37,38);(1,4,5,3)(2,8,6,7)(9,14,15,21)(10,18,13,24)(11,22,20,12)(16,23,19,17)(25,29,30,37)(26,34,31,40)(27,33,35,39)(28,32,36,38)Polycyclic presentation:
2^9|b^2=hi,c^2=gi,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7618760);