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 | |
4386 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)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26)(27,34)(28,36)(29,31)(30,32)(33,37)(35,39)(38,40);(1,2,5,4)(3,7,8,6)(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23)(25,27,31,30)(26,32,29,34)(28,35,37,38)(33,39,36,40);(2,4)(6,7)(12,20)(15,22)(18,24)(21,23)(25,28)(26,33)(27,35)(29,36)(30,38)(31,37)(32,39)(34,40);(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,26,31,29)(27,34,30,32)(28,33,37,36)(35,40,38,39);(1,3,5,8)(2,6,4,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,29,31,26)(27,32,30,34)(28,36,37,33)(35,39,38,40);(1,4,5,2)(3,7,8,6)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,30,31,27)(26,32,29,34)(28,35,37,38)(33,40,36,39)Polycyclic presentation:
2^9|b^2=gi,d^2=h,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8161747);