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 | |
4307 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,14,16)(11,15,19,22)(12,17,21,24)(13,20,23,18)(26,31)(27,34)(29,37)(36,39);(2,3)(6,7)(9,11,14,19)(10,15,16,22)(12,20,21,18)(13,17,23,24)(25,26,30,31)(27,28,34,32)(29,33,37,38)(35,36,40,39);(1,3,5,2)(4,7,8,6)(10,16)(11,19)(17,24)(18,20)(25,27,30,34)(26,32,31,28)(29,35,37,40)(33,39,38,36);(1,4)(2,6)(3,7)(5,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,23)(21,24)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,40)(38,39);(9,12,14,21)(10,17,16,24)(11,18,19,20)(13,22,23,15)(25,28)(26,27)(29,36)(30,32)(31,34)(33,35)(37,39)(38,40);(9,13)(10,18)(11,17)(12,15)(14,23)(16,20)(19,24)(21,22)(25,29,30,37)(26,33,31,38)(27,35,34,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=g,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,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, 9619687);