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 |
| prime radical | #49577 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(10,15)(11,19)(17,20)(18,23)(25,26)(27,30)(28,36)(29,33)(31,35)(32,38)(34,40)(37,39);(1,2,5,6)(3,7,4,8)(9,10)(11,16)(12,17)(13,23)(14,15)(18,24)(19,22)(20,21)(25,27)(26,30)(28,34)(29,35)(31,33)(32,39)(36,40)(37,38);(10,15)(12,21)(16,22)(18,23)(25,27,29,35)(26,31,33,30)(28,37,38,34)(32,39,36,40);(9,11)(10,16)(12,18)(13,20)(14,19)(15,22)(17,24)(21,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,40)(36,39)(37,38);(1,3,5,4)(2,7,6,8)(9,12,14,21)(10,17,15,20)(11,18,19,23)(13,16,24,22)(25,28)(26,32)(27,34)(29,38)(30,39)(31,40)(33,36)(35,37);(1,4,5,3)(2,8,6,7)(9,13,14,24)(10,18,15,23)(11,20,19,17)(12,22,21,16)(25,26)(27,31)(28,36)(29,33)(30,35)(32,38)(34,39)(37,40)Polycyclic presentation:
2^9|b^2=ghi,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7876876);