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 | |
4508 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)(11,18)(12,16)(13,21)(14,15)(17,23)(19,20)(22,24)(25,26)(27,32)(28,34)(29,31)(30,33)(35,37)(36,40)(38,39);(3,8)(5,6)(9,11,15,16)(10,12,14,18)(13,19,23,24)(17,20,21,22)(25,27)(26,32)(28,38)(29,35)(30,36)(31,37)(33,40)(34,39);(1,3)(2,5)(4,8)(6,7)(10,14)(12,18)(17,21)(20,22)(25,28,31,30)(26,33,29,34)(27,36,37,38)(32,39,35,40);(1,3,4,8)(2,6,7,5)(9,12,15,18)(10,16,14,11)(13,22,23,20)(17,19,21,24)(25,29,31,26)(27,32,37,35)(28,33,30,34)(36,40,38,39);(1,3)(2,6)(4,8)(5,7)(9,13)(10,17)(11,19)(12,20)(14,21)(15,23)(16,24)(18,22)(25,30,31,28)(26,34,29,33)(27,36,37,38)(32,39,35,40);(1,2)(3,6)(4,7)(5,8)(9,14,15,10)(11,18,16,12)(13,21,23,17)(19,22,24,20)(25,27)(26,35)(28,38)(29,32)(30,36)(31,37)(33,39)(34,40)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8841886);