Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470037 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62848 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,16,12)(11,20,24,32)(13,18,28,25)(14,19,15,26)(17,30,21,23)(22,35,39,40)(27,33,29,34)(31,37,36,38);(1,3)(2,5)(4,7)(6,8)(9,11)(10,17)(12,21)(13,22)(14,23)(15,30)(16,24)(18,31)(19,32)(20,26)(25,36)(27,37)(28,39)(29,38)(33,35)(34,40);(3,7)(5,8)(9,12,16,10)(11,21,24,17)(13,25,28,18)(14,26,15,19)(20,30,32,23)(22,36,39,31)(27,34,29,33)(35,37,40,38);(3,7)(5,8)(9,13)(10,18)(11,22)(12,25)(14,29)(15,27)(16,28)(17,31)(19,34)(20,35)(21,36)(23,38)(24,39)(26,33)(30,37)(32,40);(2,6)(5,8)(9,14,16,15)(10,19,12,26)(11,23,24,30)(13,27,28,29)(17,32,21,20)(18,33,25,34)(22,37,39,38)(31,35,36,40)Polycyclic presentation:
2^9|a^2=i,c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377242);