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 | |
4170 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)(3,5)(4,7)(6,8)(9,10,15,18)(11,16,21,12)(13,14,17,23)(19,20,24,22)(27,35)(28,37)(31,38)(32,40);(2,3)(6,7)(9,11,15,21)(10,16,18,12)(13,19,17,24)(14,20,23,22)(25,26,30,34)(27,31,35,38)(28,32,37,40)(29,33,36,39);(1,2,5,3)(4,6,8,7)(9,12,15,16)(10,11,18,21)(13,22,17,20)(14,19,23,24)(25,27)(26,31)(28,36)(29,37)(30,35)(32,39)(33,40)(34,38);(1,3,5,2)(4,7,8,6)(9,13)(10,14)(11,19)(12,22)(15,17)(16,20)(18,23)(21,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,4,5,8)(2,6,3,7)(9,12)(10,11)(13,22)(14,19)(15,16)(17,20)(18,21)(23,24)(25,28,30,37)(26,32,34,40)(27,36,35,29)(31,39,38,33);(9,14)(10,17)(11,20)(12,19)(13,18)(15,23)(16,24)(21,22)(25,29,30,36)(26,33,34,39)(27,37,35,28)(31,40,38,32)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=i,d^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8440753);