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 | |
4475 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,14,17)(11,15,18,22)(12,13,16,21)(19,20,23,24)(25,26)(27,38)(28,35)(29,34)(30,40)(31,36)(32,37)(33,39);(3,7)(5,8)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,27)(26,33)(28,36)(29,30)(31,34)(32,39)(35,40)(37,38);(1,2)(3,5)(4,6)(7,8)(11,18)(15,22)(19,23)(20,24)(25,28,32,29)(26,34,37,35)(27,36,39,30)(31,38,40,33);(1,3)(2,5)(4,7)(6,8)(9,12)(10,13)(11,19)(14,16)(15,20)(17,21)(18,23)(22,24)(25,29,32,28)(26,35,37,34)(27,36,39,30)(31,38,40,33);(1,2)(3,5)(4,6)(7,8)(9,13)(10,16)(11,20)(12,17)(14,21)(15,23)(18,24)(19,22)(25,30,32,36)(26,31,37,40)(27,29,39,28)(33,34,38,35);(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,31)(26,36)(27,34)(28,33)(29,38)(30,37)(32,40)(35,39)Polycyclic presentation:
2^9|a^2=ghi,c^2=g,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8183636);