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 | |
4461 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(25,26,29,33)(27,30,35,31)(28,32,37,40)(34,38,36,39);(1,3,5,8)(2,6,4,7)(9,10)(11,16)(12,17)(13,21)(14,15)(18,22)(19,20)(23,24)(25,27)(26,30)(28,36)(29,35)(31,33)(32,39)(34,37)(38,40);(10,15)(12,16)(18,21)(20,23)(25,27,29,35)(26,31,33,30)(28,34,37,36)(32,39,40,38);(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,23,21,20)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,37)(39,40);(1,2,5,4)(3,6,8,7)(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,39)(37,40);(1,4,5,2)(3,7,8,6)(9,13)(10,18)(11,19)(12,20)(14,22)(15,21)(16,23)(17,24)(25,28)(26,32)(27,34)(29,37)(30,38)(31,39)(33,40)(35,36)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=h,d^2=hi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7636230);