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 | #41022 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4183 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)(3,4)(5,6)(7,8)(9,10,15,13)(11,12,18,16)(14,21,22,17)(19,23,24,20)(26,29)(31,34)(32,37)(38,40);(2,6)(4,7)(10,13)(11,18)(17,21)(19,24)(28,36)(32,37)(33,39)(38,40);(2,6)(3,8)(9,11,15,18)(10,16,13,12)(14,19,22,24)(17,23,21,20)(25,26,30,29)(27,31,35,34)(28,32,36,37)(33,38,39,40);(1,3,5,8)(2,7,6,4)(9,12)(10,11)(13,18)(14,20)(15,16)(17,19)(21,24)(22,23)(25,27,30,35)(26,31,29,34)(28,33,36,39)(32,38,37,40);(1,2)(3,7)(4,8)(5,6)(9,13,15,10)(11,12,18,16)(14,21,22,17)(19,20,24,23)(25,28,30,36)(26,32,29,37)(27,33,35,39)(31,38,34,40);(1,4,5,7)(2,8,6,3)(9,14,15,22)(10,17,13,21)(11,19,18,24)(12,20,16,23)(25,29,30,26)(27,34,35,31)(28,37,36,32)(33,40,39,38)Polycyclic presentation:
2^9|a^2=i,c^2=ghi,d^2=hi,e^2=ghi,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, 8714017);