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 | #55153 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4173 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)(11,18)(12,16)(13,21)(14,15)(17,23)(19,20)(22,24)(25,26,30,33)(27,31,34,38)(28,29,32,37)(35,36,39,40);(1,3,6,4)(2,7,5,8)(10,15)(12,18)(17,21)(20,22)(27,34)(31,38)(35,39)(36,40);(10,15)(11,16)(17,21)(19,24)(25,27,30,34)(26,31,33,38)(28,35,32,39)(29,36,37,40);(1,3,6,4)(2,7,5,8)(9,11,14,16)(10,12,15,18)(13,19,23,24)(17,20,21,22);(1,4,6,3)(2,7,5,8)(9,12)(10,16)(11,15)(13,22)(14,18)(17,19)(20,23)(21,24)(25,28)(26,29)(27,35)(30,32)(31,36)(33,37)(34,39)(38,40);(1,5,6,2)(3,8,4,7)(9,13,14,23)(10,17,15,21)(11,19,16,24)(12,20,18,22)(25,29)(26,32)(27,36)(28,33)(30,37)(31,39)(34,40)(35,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=i,d^2=g,e^2=gh,f^2=g,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, 9592781);