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 | |
4492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,15)(11,17,18,16)(12,21,23,13)(19,24,22,20)(25,26)(27,28)(29,32)(30,38)(31,34)(33,40)(35,37)(36,39);(1,2,4,6)(3,5,8,7)(10,15)(13,21)(16,17)(20,24)(25,27,31,32)(26,28,34,29)(30,39,40,35)(33,37,38,36);(1,2)(3,5)(4,6)(7,8)(11,18)(16,17)(19,22)(20,24)(25,28,31,29)(26,32,34,27)(30,36,40,37)(33,39,38,35);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,22)(13,20)(14,18)(15,17)(19,23)(21,24)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40);(9,12)(10,13)(11,19)(14,23)(15,21)(16,24)(17,20)(18,22)(25,29,31,28)(26,27,34,32)(30,37,40,36)(33,35,38,39);(9,13)(10,12)(11,20)(14,21)(15,23)(16,22)(17,19)(18,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,40)(32,39)(34,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=h,d^2=hi,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, 8751758);