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 | |
4486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,4)(6,7)(9,10)(11,12)(13,18)(14,15)(16,17)(19,20)(21,24)(22,23)(25,26)(27,30)(28,33)(29,32)(31,36)(34,37)(35,39)(38,40);(1,2,5,8)(3,6,4,7)(10,15)(11,17)(18,23)(19,21)(26,27)(31,33)(32,34)(38,39);(1,3,5,4)(2,6,8,7)(10,15)(12,16)(18,23)(20,24)(25,26,30,27)(28,31,36,33)(29,32,37,34)(35,38,40,39);(1,2)(3,6)(4,7)(5,8)(9,11,14,17)(10,16,15,12)(13,21,22,19)(18,20,23,24)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(1,4,5,3)(2,7,8,6)(9,12,14,16)(10,17,15,11)(13,20,22,24)(18,21,23,19)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(9,13,14,22)(10,18,15,23)(11,19,17,21)(12,20,16,24)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)Polycyclic presentation:
2^9|b^2=gh,c^2=ghi,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8442307);