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 | |
4442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,3)(4,8)(9,10,14,16)(11,15,17,24)(12,20,22,13)(18,19,21,23)(25,26,30,33)(27,34,31,28)(29,32,38,40)(35,39,37,36);(2,3)(4,8)(9,11)(10,15)(12,18)(13,23)(14,17)(16,24)(19,20)(21,22)(25,27,30,31)(26,28,33,34)(29,37,38,35)(32,39,40,36);(1,2,5,3)(4,6,8,7)(11,17)(15,24)(18,21)(19,23)(25,26,30,33)(27,28,31,34)(29,32,38,40)(35,36,37,39);(2,3)(6,7)(9,11)(10,15)(12,21)(13,19)(14,17)(16,24)(18,22)(20,23)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40);(1,3,5,2)(4,7,8,6)(9,12)(10,13)(11,18)(14,22)(15,23)(16,20)(17,21)(19,24)(25,28)(26,31)(27,33)(29,36)(30,34)(32,37)(35,40)(38,39);(1,4)(2,6)(3,7)(5,8)(9,13)(10,12)(11,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,37)(33,40)(34,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8733492);