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 | |
4640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,7)(3,6,8,5)(9,10)(11,13)(12,16)(14,15)(17,21)(18,20)(19,22)(23,24)(25,26,32,34)(27,35,31,33)(28,30,29,39)(36,40,37,38);(3,8)(5,6)(12,19)(16,22)(18,23)(20,24)(25,27,32,31)(26,33,34,35)(28,36,29,37)(30,38,39,40);(1,3)(2,5)(4,8)(6,7)(10,15)(11,17)(16,22)(18,23)(25,28,32,29)(26,30,34,39)(27,36,31,37)(33,38,35,40);(3,8)(5,6)(9,11,14,17)(10,13,15,21)(12,18,19,23)(16,20,22,24)(25,29,32,28)(26,30,34,39)(27,37,31,36)(33,38,35,40);(1,3)(2,6)(4,8)(5,7)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,30)(26,29)(27,38)(28,34)(31,40)(32,39)(33,37)(35,36);(1,2)(3,5)(4,7)(6,8)(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,31,32,27)(26,33,34,35)(28,37,29,36)(30,38,39,40)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=g,d^2=ghi,f^2=g,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, 10028358);