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 | |
4410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(9,10)(11,18)(12,15)(13,20)(14,16)(17,22)(19,23)(21,24)(25,26)(27,32)(28,35)(29,34)(30,33)(31,36)(37,39)(38,40);(1,2,5,3)(4,6,8,7)(9,11,15,14)(10,16,12,18)(13,19,22,21)(17,23,20,24)(25,27)(26,29)(28,37)(30,38)(31,34)(32,36)(33,39)(35,40);(2,3)(6,7)(9,10,15,12)(11,18,14,16)(13,17,22,20)(19,24,21,23)(25,27)(26,32)(28,37)(29,36)(30,38)(31,34)(33,40)(35,39);(9,12,15,10)(11,16,14,18)(13,20,22,17)(19,23,21,24)(25,28,31,30)(26,33,36,35)(27,37,34,38)(29,39,32,40);(1,3,5,2)(4,7,8,6)(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,29,31,32)(26,34,36,27)(28,39,30,40)(33,38,35,37);(1,4)(2,6)(3,7)(5,8)(9,14,15,11)(10,16,12,18)(13,21,22,19)(17,23,20,24)(25,30,31,28)(26,35,36,33)(27,38,34,37)(29,40,32,39)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8027202);