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 | |
4512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,19)(12,20)(13,18)(14,16)(15,21)(17,23)(22,24)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,39)(36,40);(1,2)(3,7)(4,5)(6,8)(9,11,14,21)(10,15,16,19)(12,18,23,22)(13,20,24,17)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,39)(36,40);(2,5)(6,7)(10,16)(13,24)(15,19)(17,20)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(1,3,4,8)(2,6,5,7)(9,12)(10,17)(11,18)(13,15)(14,23)(16,20)(19,24)(21,22)(25,28,30,27)(26,31,33,34)(29,35,38,36)(32,40,37,39);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,22)(13,20)(14,21)(16,19)(17,24)(18,23)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,24)(10,18,16,22)(11,20,21,17)(12,15,23,19)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8298101);