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 | |
4462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,22)(23,24)(25,26,31,36)(27,34,38,32)(28,33,29,39)(30,35,40,37);(2,5)(6,7)(10,14)(15,17)(16,20)(22,23)(25,27)(26,32)(28,35)(29,37)(30,33)(31,38)(34,36)(39,40);(1,2)(3,6)(4,5)(7,8)(10,14)(12,21)(15,17)(18,24)(25,28,31,29)(26,33,36,39)(27,35,38,37)(30,34,40,32);(1,3,4,8)(2,6,5,7)(9,11,13,19)(10,15,14,17)(12,18,21,24)(16,22,20,23)(25,29,31,28)(26,33,36,39)(27,37,38,35)(30,34,40,32);(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27)(26,34)(28,37)(29,35)(30,33)(31,38)(32,36)(39,40);(9,12,13,21)(10,16,14,20)(11,18,19,24)(15,22,17,23)(25,30)(26,35)(27,33)(28,32)(29,34)(31,40)(36,37)(38,39)Polycyclic presentation:
2^9|a^2=ghi,c^2=ghi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8851619);