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 | #39120 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,7)(3,5,4,8)(9,10)(11,18)(12,13)(14,15)(16,19)(17,22)(20,21)(23,24)(25,26)(27,28)(29,37)(30,32)(31,34)(33,40)(35,39)(36,38);(2,7)(3,4)(10,15)(11,19)(12,22)(21,23)(25,27,31,32)(26,28,34,30)(29,38,40,35)(33,39,37,36);(1,3,6,4)(2,8,7,5)(10,15)(12,22)(16,18)(20,24)(25,28,31,30)(26,32,34,27)(29,39,40,36)(33,35,37,38);(1,4,6,3)(2,5,7,8)(9,11)(10,16)(12,20)(13,23)(14,19)(15,18)(17,21)(22,24)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,39)(38,40);(1,5)(2,3)(4,7)(6,8)(9,12)(10,17)(11,20)(13,15)(14,22)(16,21)(18,23)(19,24)(25,29,31,40)(26,33,34,37)(27,35,32,38)(28,36,30,39);(9,13,14,17)(10,12,15,22)(11,21,19,23)(16,24,18,20)(25,30,31,28)(26,32,34,27)(29,39,40,36)(33,38,37,35)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=i,d^2=ghi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8556182);