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 | |
4457 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,20)(13,17)(15,19)(16,22)(18,24)(21,23)(25,26,30,33)(27,31,35,28)(29,32,38,40)(34,39,37,36);(2,6)(3,8)(9,11,13,19)(10,14,17,15)(12,21,22,18)(16,24,20,23)(25,27)(26,31)(28,33)(29,37)(30,35)(32,36)(34,38)(39,40);(1,3)(2,4)(5,8)(6,7)(9,10,13,17)(11,15,19,14)(12,16,22,20)(18,24,21,23)(25,26,30,33)(27,28,35,31)(29,32,38,40)(34,36,37,39);(1,4)(2,3)(5,7)(6,8)(9,10)(11,14)(12,16)(13,17)(15,19)(18,23)(20,22)(21,24)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,40,37);(1,2)(3,4)(5,6)(7,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,24)(20,23)(21,22)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,38)(39,40);(1,4)(2,3)(5,7)(6,8)(9,12)(10,16)(11,18)(13,22)(14,23)(15,24)(17,20)(19,21)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=gi,d^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8757779);