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 | |
4666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,14,15)(11,16,21,24)(12,17,22,20)(13,18,23,19)(25,26,31,35)(27,36,37,32)(28,30,34,40)(29,39,38,33);(1,2)(3,6)(4,7)(5,8)(25,27)(26,32)(28,38)(29,34)(30,39)(31,37)(33,40)(35,36);(1,3,5,4)(2,6,8,7)(10,15)(13,23)(16,24)(17,20)(27,37)(29,38)(32,36)(33,39);(9,11)(10,16)(12,18)(13,20)(14,21)(15,24)(17,23)(19,22)(25,28)(26,30)(27,38)(29,37)(31,34)(32,39)(33,36)(35,40);(1,4,5,3)(2,7,8,6)(9,12)(10,17)(11,19)(13,16)(14,22)(15,20)(18,21)(23,24)(25,29,31,38)(26,33,35,39)(27,34,37,28)(30,32,40,36);(9,13)(10,18)(11,20)(12,16)(14,23)(15,19)(17,21)(22,24)(25,30)(26,34)(27,39)(28,35)(29,32)(31,40)(33,37)(36,38)Polycyclic presentation:
2^9|a^2=gh,c^2=gi,e^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7982393);