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 | |
4324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26)(27,35)(28,32)(29,33)(30,34)(31,36)(37,38)(39,40);(1,3,5,8)(2,6,4,7)(11,17)(15,21)(18,23)(22,24)(25,27,30,36)(26,31,34,35)(28,29,37,39)(32,40,38,33);(1,2,5,4)(3,6,8,7)(10,14)(15,21)(16,19)(22,24)(27,36)(29,39)(31,35)(33,40);(1,2,5,4)(3,6,8,7)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,28,30,37)(26,32,34,38)(27,29,36,39)(31,40,35,33);(1,4,5,2)(3,7,8,6)(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,26)(27,31)(28,38)(29,33)(30,34)(32,37)(35,36)(39,40);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29)(26,33)(27,37)(28,36)(30,39)(31,32)(34,40)(35,38)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=gi,d^2=g,e^2=ghi,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7921915);