Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7501009 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1934 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)(9,10,15,21,14,20,13,19)(11,22,24,33,25,16,26,18)(12,27,29,17,30,35,31,36)(23,38,40,37,39,34,28,32);(1,3,5,2)(4,7,8,6)(9,11,14,25)(10,16,20,22)(12,28,30,40)(13,26,15,24)(17,34,36,38)(18,19,33,21)(23,31,39,29)(27,37,35,32);(1,4,5,8)(2,6,3,7)(9,12,14,30)(10,17,20,36)(11,23,25,39)(13,29,15,31)(16,32,22,37)(18,34,33,38)(19,35,21,27)(24,28,26,40);(2,3)(6,7)(9,11)(10,18)(12,23)(13,24)(14,25)(15,26)(16,21)(17,34)(19,22)(20,33)(27,32)(28,29)(30,39)(31,40)(35,37)(36,38);(9,13,14,15)(10,19,20,21)(11,24,25,26)(12,29,30,31)(16,18,22,33)(17,35,36,27)(23,28,39,40)(32,34,37,38)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,h^b=hi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410917);