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 | #51784 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4211 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,8,4,7)(9,10,14,15)(11,19,20,16)(12,17,21,24)(13,22,23,18)(25,26)(27,30)(28,36)(29,32)(31,38)(33,35)(34,40)(37,39);(1,3,6,4)(2,7,5,8)(10,15)(12,21)(16,19)(18,22)(25,26)(27,33)(28,31)(29,32)(30,35)(34,40)(36,38)(37,39);(10,15)(11,20)(17,24)(18,22)(25,26,29,32)(27,30,35,33)(28,31,38,36)(34,39,37,40);(1,3,6,4)(2,7,5,8)(9,11,14,20)(10,16,15,19)(12,18,21,22)(13,17,23,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,39)(36,38)(37,40);(1,4,6,3)(2,7,5,8)(9,12)(10,17)(11,18)(13,19)(14,21)(15,24)(16,23)(20,22)(25,27,29,35)(26,30,32,33)(28,37,38,34)(31,40,36,39);(1,5,6,2)(3,8,4,7)(9,13,14,23)(10,18,15,22)(11,17,20,24)(12,16,21,19)(25,28,29,38)(26,31,32,36)(27,34,35,37)(30,39,33,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=i,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9619263);