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 | |
4397 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,26)(27,38)(28,34)(29,39)(30,35)(31,36)(32,37)(33,40);(1,2)(3,7)(4,6)(5,8)(12,19)(16,22)(18,23)(20,24)(25,27,32,40)(26,33,37,38)(28,30,29,36)(31,39,35,34);(1,3)(2,5)(4,8)(6,7)(10,15)(11,17)(16,22)(18,23)(25,28,32,29)(26,34,37,39)(27,30,40,36)(31,38,35,33);(2,6)(5,7)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,29,32,28)(26,34,37,39)(27,30,40,36)(31,33,35,38);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,30)(26,35)(27,29)(28,40)(31,37)(32,36)(33,34)(38,39);(1,3)(2,7)(4,8)(5,6)(9,13)(10,11)(12,20)(14,21)(15,17)(16,18)(19,24)(22,23)(25,31,32,35)(26,36,37,30)(27,39,40,34)(28,38,29,33)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10125012);