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 | |
4441 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)(3,4)(5,6)(7,8)(25,26)(27,30)(28,31)(29,33)(32,35)(34,38)(36,37)(39,40);(2,6)(4,7)(9,10,14,12)(11,15,17,18)(13,20,21,16)(19,24,23,22)(25,27,29,35)(26,30,33,32)(28,34,37,40)(31,38,36,39);(2,6)(3,8)(10,12)(11,17)(16,20)(19,23)(25,26,29,33)(27,32,35,30)(28,36,37,31)(34,38,40,39);(1,3,5,8)(2,7,6,4)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28,29,37)(26,31,33,36)(27,34,35,40)(30,38,32,39);(1,2)(3,7)(4,8)(5,6)(9,12,14,10)(11,18,17,15)(13,20,21,16)(19,24,23,22)(25,26)(27,32)(28,31)(29,33)(30,35)(34,39)(36,37)(38,40);(1,4,5,7)(2,8,6,3)(9,13,14,21)(10,16,12,20)(11,19,17,23)(15,22,18,24)(25,27)(26,32)(28,34)(29,35)(30,33)(31,39)(36,38)(37,40)Polycyclic presentation:
2^9|b^2=i,c^2=gh,d^2=h,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8729009);