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 | |
4337 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(25,26)(27,36)(28,30)(29,33)(31,35)(32,37)(34,39)(38,40);(2,5)(6,7)(9,10,15,13)(11,18,12,16)(14,17,22,21)(19,24,20,23)(27,37)(29,38)(32,36)(33,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,16,13,18)(14,19,22,20)(17,23,21,24)(25,27,31,37)(26,32,35,36)(28,38,34,29)(30,33,39,40);(1,3)(2,6)(4,8)(5,7)(9,12,15,11)(10,16,13,18)(14,20,22,19)(17,23,21,24)(25,28,31,34)(26,30,35,39)(27,29,37,38)(32,40,36,33);(9,13,15,10)(11,18,12,16)(14,17,22,21)(19,23,20,24)(25,29)(26,33)(27,28)(30,36)(31,38)(32,39)(34,37)(35,40);(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,30)(26,34)(27,33)(28,35)(29,32)(31,39)(36,38)(37,40)Polycyclic presentation:
2^9|b^2=ghi,c^2=gi,d^2=gi,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10088394);