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 | |
4546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,13)(11,12,19,16)(14,17,22,21)(18,20,24,23)(27,35)(29,39)(31,36)(33,40);(1,2)(3,7)(4,5)(6,8)(9,11,15,19)(10,16,13,12)(14,18,22,24)(17,23,21,20)(25,26)(27,36)(28,38)(29,40)(30,34)(31,35)(32,37)(33,39);(2,5)(6,7)(14,22)(17,21)(18,24)(20,23)(25,27,30,35)(26,31,34,36)(28,29,37,39)(32,40,38,33);(3,8)(6,7)(9,12)(10,11)(13,19)(14,20)(15,16)(17,18)(21,24)(22,23)(25,28)(26,32)(27,29)(30,37)(31,40)(33,36)(34,38)(35,39);(1,3)(2,6)(4,8)(5,7)(9,13,15,10)(11,16,19,12)(14,21,22,17)(18,23,24,20)(25,29)(26,33)(27,37)(28,35)(30,39)(31,32)(34,40)(36,38);(1,2)(3,6)(4,5)(7,8)(9,14)(10,17)(11,18)(12,20)(13,21)(15,22)(16,23)(19,24)(25,26)(27,36)(28,32)(29,33)(30,34)(31,35)(37,38)(39,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,e^2=g,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, 8253879);