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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,8)(5,6)(9,10,13,15)(11,16,19,14)(12,20,21,17)(18,22,24,23)(25,26,31,36)(27,32,29,34)(28,38,37,33)(30,35,39,40);(1,2,5,6)(3,7,8,4)(9,11,13,19)(10,14,15,16)(12,18,21,24)(17,22,20,23)(25,27,31,29)(26,32,36,34)(28,39,37,30)(33,35,38,40);(1,3)(2,4)(5,8)(6,7)(10,15)(12,21)(14,16)(18,24)(25,28,31,37)(26,33,36,38)(27,30,29,39)(32,40,34,35);(1,4)(2,3)(5,7)(6,8)(9,10)(11,14)(12,17)(13,15)(16,19)(18,22)(20,21)(23,24)(25,29,31,27)(26,34,36,32)(28,30,37,39)(33,40,38,35);(1,2)(3,4)(5,6)(7,8)(9,11)(10,16)(12,18)(13,19)(14,15)(17,23)(20,22)(21,24)(25,30)(26,35)(27,37)(28,29)(31,39)(32,33)(34,38)(36,40);(1,4)(2,3)(5,7)(6,8)(9,12)(10,17)(11,18)(13,21)(14,22)(15,20)(16,23)(19,24)(25,26)(27,32)(28,38)(29,34)(30,35)(31,36)(33,37)(39,40)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=gh,d^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8812433);