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 | |
4494 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)(25,26,30,29)(27,33,35,31)(28,32,37,38)(34,40,36,39);(1,2,5,6)(3,7,8,4)(10,14)(11,17)(16,20)(19,21)(25,27,30,35)(26,31,29,33)(28,34,37,36)(32,39,38,40);(1,3)(2,4)(5,8)(6,7)(10,14)(15,18)(16,20)(23,24);(2,6)(4,7)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,22)(21,23)(25,27)(26,31)(28,36)(29,33)(30,35)(32,40)(34,37)(38,39);(1,4)(2,3)(5,7)(6,8)(9,11)(10,15)(12,21)(13,17)(14,18)(16,24)(19,22)(20,23)(25,28)(26,32)(27,34)(29,38)(30,37)(31,39)(33,40)(35,36);(1,3)(2,7)(4,6)(5,8)(9,12,13,22)(10,16,14,20)(11,19,17,21)(15,23,18,24)(25,29,30,26)(27,33,35,31)(28,32,37,38)(34,39,36,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,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, 10132739);