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 | |
4510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,13,14)(11,15,17,18)(12,20,22,16)(19,24,21,23)(25,26)(27,31)(28,35)(29,30)(32,36)(33,37)(34,40)(38,39);(1,2,5,6)(3,7,8,4)(10,14)(11,17)(16,20)(19,21)(27,33)(31,37)(34,39)(38,40);(1,3)(2,4)(5,8)(6,7)(10,14)(15,18)(16,20)(23,24)(26,30)(31,37)(32,35)(38,40);(2,6)(4,7)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,22)(21,23)(25,26)(27,31)(28,32)(29,30)(33,37)(34,38)(35,36)(39,40);(1,4)(2,3)(5,7)(6,8)(9,11,13,17)(10,15,14,18)(12,21,22,19)(16,24,20,23)(25,27)(26,31)(28,34)(29,33)(30,37)(32,38)(35,40)(36,39);(1,3)(2,7)(4,6)(5,8)(9,12,13,22)(10,16,14,20)(11,19,17,21)(15,23,18,24)(25,28)(26,32)(27,34)(29,36)(30,35)(31,38)(33,39)(37,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,e^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, 10132639);