Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1423 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,7,4,6)(25,26)(27,32)(28,30)(29,31)(33,36)(34,37)(35,38)(39,40);(1,3,5,4)(2,6,8,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(26,30)(27,34)(31,33)(35,40);(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(26,31)(27,34)(29,36)(30,33)(32,39)(37,38);(1,3,5,4)(2,7,8,6)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,27)(26,32)(28,34)(29,35)(30,37)(31,38)(33,39)(36,40);(1,4,5,3)(2,6,8,7)(10,16)(14,17)(15,24)(22,23)(27,34)(32,37)(35,40)(38,39)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=h,d^2=fhi,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6438137);