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 | |
4673 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)(3,5)(4,7)(6,8)(9,10)(11,16)(12,22)(13,17)(14,18)(15,19)(20,24)(21,23)(25,26)(27,31)(28,37)(29,36)(30,32)(33,38)(34,40)(35,39);(2,3)(6,7)(9,11,15,20)(10,16,19,24)(12,23,17,14)(13,18,22,21)(25,27,30,35)(26,31,32,39)(28,36,38,34)(29,33,40,37);(1,2,5,3)(4,6,8,7)(11,20)(14,23)(16,24)(18,21)(26,32)(27,35)(33,37)(34,36);(1,3,5,2)(4,7,8,6)(9,12)(10,13)(11,14)(15,17)(16,21)(18,24)(19,22)(20,23)(25,28)(26,33)(27,36)(29,39)(30,38)(31,40)(32,37)(34,35);(1,4)(2,6)(3,7)(5,8)(9,13)(10,17)(11,21)(12,19)(14,24)(15,22)(16,23)(18,20)(25,29,30,40)(26,34,32,36)(27,37,35,33)(28,31,38,39);(9,14)(10,18)(11,12)(13,24)(15,23)(16,22)(17,20)(19,21)(25,26)(27,31)(28,33)(29,34)(30,32)(35,39)(36,40)(37,38)Polycyclic presentation:
2^9|b^2=gi,c^2=g,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8314321);