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 | |
4495 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,15)(13,17)(14,16)(19,24)(20,22)(21,23)(25,26)(27,32)(28,30)(29,34)(31,33)(35,40)(36,37)(38,39);(1,2)(3,8)(4,6)(5,7)(9,11,15,14)(10,16,12,18)(13,21,22,19)(17,24,20,23)(25,27)(26,32)(28,36)(29,33)(30,37)(31,34)(35,39)(38,40);(1,3,5,4)(2,6,7,8)(10,12)(16,18)(17,20)(23,24)(26,33)(27,34)(28,35)(37,38);(9,12,15,10)(11,18,14,16)(13,20,22,17)(19,24,21,23)(25,28,31,35)(26,30,33,40)(27,36,34,39)(29,38,32,37);(1,4,5,3)(2,6,7,8)(9,13,15,22)(10,17,12,20)(11,19,14,21)(16,23,18,24)(25,29,31,32)(26,34,33,27)(28,38,35,37)(30,39,40,36);(1,2)(3,6)(4,8)(5,7)(9,14,15,11)(10,18,12,16)(13,19,22,21)(17,23,20,24)(25,30)(26,35)(27,37)(28,33)(29,36)(31,40)(32,39)(34,38)Polycyclic presentation:
2^9|b^2=g,c^2=ghi,d^2=h,e^2=gi,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, 9613848);