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 | #39445 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4168 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)(11,19)(12,14)(13,16)(15,18)(17,23)(20,24)(21,22)(25,26,32,36)(27,35,31,33)(28,29,30,34)(37,40,39,38);(1,2)(3,8)(4,6)(5,7)(9,11,15,13)(10,16,18,19)(12,20,23,22)(14,21,17,24)(25,27,32,31)(26,33,36,35)(28,37,30,39)(29,38,34,40);(1,3,5,4)(2,6,7,8)(25,28,32,30)(26,34,36,29)(27,37,31,39)(33,40,35,38);(9,12)(10,17)(11,20)(13,22)(14,18)(15,23)(16,24)(19,21)(25,29,32,34)(26,28,36,30)(27,38,31,40)(33,37,35,39);(1,4,5,3)(2,6,7,8)(9,13,15,11)(10,16,18,19)(12,22,23,20)(14,21,17,24)(25,30,32,28)(26,34,36,29)(27,37,31,39)(33,38,35,40);(1,2)(3,6)(4,8)(5,7)(9,14,15,17)(10,12,18,23)(11,21,13,24)(16,20,19,22)(25,31,32,27)(26,35,36,33)(28,39,30,37)(29,40,34,38)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=hi,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9613870);