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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4130 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)(11,12)(13,16)(14,21)(15,17)(18,23)(19,24)(20,22)(25,26,31,28)(27,34,30,32)(29,36,38,33)(35,39,37,40);(1,2)(3,7)(4,5)(6,8)(9,11,16,15)(10,17,13,12)(14,19,23,22)(18,24,21,20)(25,27,31,30)(26,32,28,34)(29,35,38,37)(33,39,36,40);(2,5)(6,7)(9,12)(10,15)(11,13)(14,20)(16,17)(18,22)(19,21)(23,24)(25,28,31,26)(27,32,30,34)(29,36,38,33)(35,39,37,40);(9,13,16,10)(11,17,15,12)(14,21,23,18)(19,24,22,20)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40);(1,3,4,8)(2,6,5,7)(9,14)(10,18)(11,19)(12,20)(13,21)(15,22)(16,23)(17,24)(25,27,31,30)(26,34,28,32)(29,35,38,37)(33,40,36,39);(1,2)(3,6)(4,5)(7,8)(9,15,16,11)(10,17,13,12)(14,22,23,19)(18,24,21,20)(25,30,31,27)(26,32,28,34)(29,37,38,35)(33,39,36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gh,d^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8122072);