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 | #47863 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,20)(12,18)(13,19)(14,16)(15,17)(21,22)(23,24)(25,26)(27,36)(28,32)(29,33)(30,34)(31,35)(37,38)(39,40);(2,3)(6,7)(9,11,15,14)(10,16,17,20)(12,19,22,23)(13,18,24,21)(25,27,31,34)(26,30,35,36)(28,37,33,40)(29,38,32,39);(1,3,5,2)(4,7,8,6)(10,17)(12,22)(16,20)(19,23)(25,28,31,33)(26,32,35,29)(27,37,34,40)(30,39,36,38);(1,4)(2,6)(3,7)(5,8)(9,12,15,22)(10,18,17,21)(11,19,14,23)(13,16,24,20)(25,28)(26,32)(27,37)(29,35)(30,39)(31,33)(34,40)(36,38);(9,13,15,24)(10,19,17,23)(11,21,14,18)(12,20,22,16)(25,29,31,32)(26,33,35,28)(27,38,34,39)(30,40,36,37);(9,14,15,11)(10,20,17,16)(12,23,22,19)(13,21,24,18)(25,30)(26,34)(27,35)(28,39)(29,37)(31,36)(32,40)(33,38)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=gh,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9113611);