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 | |
4636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,19)(12,17)(13,18)(14,15)(16,20)(21,22)(23,24)(25,26)(27,30)(28,35)(29,31)(32,37)(33,39)(34,40)(36,38);(1,2)(3,8)(4,5)(6,7)(10,15)(13,23)(16,19)(17,21)(25,27,29,33)(26,30,31,39)(28,36,37,34)(32,40,35,38);(2,7)(3,4)(11,20)(12,22)(16,19)(17,21)(26,31)(27,33)(32,35)(34,36);(1,3,6,4)(2,8,7,5)(9,11)(10,16)(12,18)(13,21)(14,20)(15,19)(17,23)(22,24)(25,26)(27,30)(28,32)(29,31)(33,39)(34,38)(35,37)(36,40);(1,4,6,3)(2,5,7,8)(9,12)(10,17)(11,18)(13,19)(14,22)(15,21)(16,23)(20,24)(25,27)(26,30)(28,34)(29,33)(31,39)(32,38)(35,40)(36,37);(1,5)(2,3)(4,7)(6,8)(9,13,14,23)(10,18,15,24)(11,21,20,17)(12,19,22,16)(25,28)(26,32)(27,34)(29,37)(30,38)(31,35)(33,36)(39,40)Polycyclic presentation:
2^9|b^2=h,d^2=hi,e^2=hi,f^2=g,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, 8705909);