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 | |
4791 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(11,17)(15,18)(19,23)(22,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,37)(36,39)(38,40);(1,2,4,6)(3,5,7,8)(9,10,13,14)(11,15,17,18)(12,16,20,21)(19,22,23,24)(26,31)(28,37)(32,39)(34,38);(1,3)(2,5)(4,7)(6,8)(10,14)(11,17)(12,20)(22,24)(25,27,30,36)(26,32,31,39)(28,38,37,34)(29,33,40,35);(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,27)(26,32)(28,38)(29,35)(30,36)(31,39)(33,40)(34,37);(1,2)(3,5)(4,6)(7,8)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,28,30,37)(26,33,31,35)(27,34,36,38)(29,39,40,32);(1,3)(2,5)(4,7)(6,8)(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,29)(26,34)(27,35)(28,39)(30,40)(31,38)(32,37)(33,36)Polycyclic presentation:
2^9|b^2=hi,c^2=g,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=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, 8957605);