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 | |
4397 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)(11,12)(13,21)(14,15)(16,22)(17,18)(19,24)(20,23)(25,26)(27,29)(28,37)(30,38)(31,33)(32,36)(34,35)(39,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,19)(14,17)(15,18)(16,20)(21,24)(22,23)(25,27,31,36)(26,32,33,29)(28,34,38,40)(30,35,37,39);(1,2,4,6)(3,7,8,5)(10,15)(11,17)(13,22)(20,24)(26,33)(29,32)(30,37)(35,39);(3,8)(5,7)(9,12)(10,11)(13,20)(14,18)(15,17)(16,19)(21,23)(22,24)(25,28)(26,30)(27,34)(29,39)(31,38)(32,35)(33,37)(36,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,18)(12,17)(13,16)(14,15)(19,24)(20,23)(21,22)(25,29,31,32)(26,27,33,36)(28,35,38,39)(30,40,37,34);(1,2)(3,5)(4,6)(7,8)(9,13)(10,16)(11,19)(12,20)(14,22)(15,21)(17,23)(18,24)(25,30)(26,28)(27,35)(29,40)(31,37)(32,34)(33,38)(36,39)Polycyclic presentation:
2^9|b^2=g,c^2=ghi,e^2=g,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, 8849995);