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 | #42907 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4279 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)(9,10)(11,18)(12,20)(13,15)(14,17)(16,22)(19,23)(21,24)(25,26)(27,34)(28,37)(29,31)(30,33)(32,36)(35,39)(38,40);(1,3,5,8)(2,6,4,7)(9,11,14,13)(10,15,17,18)(12,19,22,21)(16,23,20,24)(25,27)(26,32)(28,38)(29,34)(30,35)(31,36)(33,40)(37,39);(9,10)(11,18)(12,16)(13,15)(14,17)(19,24)(20,22)(21,23)(25,28,31,30)(26,33,29,37)(27,35,36,38)(32,39,34,40);(1,2,5,4)(3,6,8,7)(9,12,14,22)(10,16,17,20)(11,19,13,21)(15,23,18,24)(25,29,31,26)(27,34,36,32)(28,33,30,37)(35,39,38,40);(1,4,5,2)(3,7,8,6)(9,10)(11,15)(12,16)(13,18)(14,17)(19,23)(20,22)(21,24)(25,27)(26,32)(28,35)(29,34)(30,38)(31,36)(33,39)(37,40);(9,13,14,11)(10,15,17,18)(12,21,22,19)(16,23,20,24)(25,30,31,28)(26,33,29,37)(27,35,36,38)(32,40,34,39)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=h,d^2=g,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7834864);