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 | |
4553 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,14,12,16)(13,18,22,24)(19,21,20,23)(25,26)(27,32)(28,30)(29,33)(31,35)(34,40)(36,38)(37,39);(1,2)(3,8)(4,6)(5,7)(9,11,15,12)(10,16,17,14)(13,20,22,19)(18,21,24,23)(25,27)(26,31)(28,32)(29,34)(30,35)(33,39)(36,40)(37,38);(1,3,5,6)(2,4,7,8)(10,17)(11,12)(18,24)(19,20);(1,4)(2,6)(3,7)(5,8)(9,12,15,11)(10,14,17,16)(13,20,22,19)(18,21,24,23)(25,28,30,26)(27,32,35,31)(29,36,38,33)(34,40,37,39);(1,2)(3,4)(5,7)(6,8)(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,27)(26,32)(28,31)(29,37)(30,35)(33,39)(34,38)(36,40);(1,4)(2,6)(3,7)(5,8)(9,14)(10,11)(12,17)(13,21)(15,16)(18,19)(20,24)(22,23)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,40)(35,37)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=hi,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10009885);