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 | #55411 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,14)(11,17)(12,20)(21,24)(25,26,31,35)(27,29,33,40)(28,34,30,32)(36,39,37,38);(1,2)(3,8)(4,5)(6,7)(9,10,15,14)(11,18,17,13)(12,16,20,22)(19,24,23,21)(25,27)(26,29)(28,36)(30,37)(31,33)(32,38)(34,39)(35,40);(2,7)(3,4)(9,11)(10,13)(12,19)(14,18)(15,17)(16,21)(20,23)(22,24)(25,28,31,30)(26,32,35,34)(27,36,33,37)(29,38,40,39);(1,3,6,4)(2,8,7,5)(9,12)(10,16)(11,19)(13,21)(14,22)(15,20)(17,23)(18,24);(1,4,6,3)(2,5,7,8)(9,13,15,18)(10,17,14,11)(12,21,20,24)(16,23,22,19)(25,29)(26,33)(27,35)(28,38)(30,39)(31,40)(32,37)(34,36);(1,5)(2,3)(4,7)(6,8)(9,14,15,10)(11,18,17,13)(12,22,20,16)(19,24,23,21)(25,30,31,28)(26,34,35,32)(27,37,33,36)(29,39,40,38)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gh,d^2=gi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,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, 8706348);