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 | |
4534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,14)(15,17)(16,21)(22,23)(25,26)(27,32)(28,34)(29,30)(31,33)(35,40)(36,39)(37,38);(1,2,5,6)(3,7,8,4)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27)(26,32)(28,30)(29,34)(31,36)(33,39)(35,38)(37,40);(2,6)(4,7)(12,20)(16,21)(18,24)(22,23)(26,33)(27,36)(30,35)(34,37);(1,3,5,8)(2,7,6,4)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,28)(26,34)(27,35)(29,39)(30,36)(31,38)(32,40)(33,37);(1,3)(2,4)(5,8)(6,7)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,29)(26,35)(27,37)(28,32)(30,33)(31,40)(34,36)(38,39);(1,4)(2,3)(5,7)(6,8)(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,35)(26,29,33,40)(27,28,36,38)(32,34,39,37)Polycyclic presentation:
2^9|b^2=hi,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8266375);