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 | |
4343 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(25,26,31,35)(27,28,36,32)(29,33,37,40)(30,34,39,38);(2,3)(6,7)(9,10,14,15)(11,12,16,19)(13,17,21,22)(18,20,23,24)(25,27)(26,32)(28,35)(29,30)(31,36)(33,38)(34,40)(37,39);(2,3)(4,8)(10,15)(12,19)(13,21)(18,23)(29,37)(30,39)(33,40)(34,38);(2,3)(6,7)(9,11)(10,12)(13,18)(14,16)(15,19)(17,20)(21,23)(22,24)(25,28,31,32)(26,27,35,36)(29,38,37,34)(30,33,39,40);(1,3,5,2)(4,7,8,6)(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,22)(21,24)(25,29)(26,33)(27,30)(28,34)(31,37)(32,38)(35,40)(36,39);(1,4,5,8)(2,6,3,7)(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,30)(26,34)(27,29)(28,33)(31,39)(32,40)(35,38)(36,37)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=gh,e^2=h,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, 8849416);