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 | #43002 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4586 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)(3,5)(4,7)(6,8)(9,10,14,15)(11,19,20,16)(12,22,23,17)(13,18,24,21)(25,26)(27,28)(29,36)(30,33)(31,32)(34,35)(37,38)(39,40);(2,3)(6,7)(10,15)(11,20)(17,22)(18,21)(25,27,31,29)(26,28,32,36)(30,34,39,37)(33,35,40,38);(1,2,5,3)(4,6,8,7)(10,15)(12,23)(16,19)(18,21)(26,32)(27,29)(33,40)(34,37);(1,3,5,2)(4,7,8,6)(9,11,14,20)(10,16,15,19)(12,21,23,18)(13,22,24,17)(25,28)(26,27)(29,32)(30,38)(31,36)(33,37)(34,40)(35,39);(1,4,5,8)(2,6,3,7)(9,12,14,23)(10,17,15,22)(11,21,20,18)(13,19,24,16)(25,29,31,27)(26,28,32,36)(30,37,39,34)(33,35,40,38);(9,13)(10,18)(11,17)(12,19)(14,24)(15,21)(16,23)(20,22)(25,30)(26,33)(27,34)(28,35)(29,37)(31,39)(32,40)(36,38)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=ghi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8436163);