Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996799 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2217 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,20,16,21,15,33)(11,22,26,36,27,35,25,17)(12,28,29,18,24,37,30,38)(13,19,23,39,32,40,31,34);(1,2)(3,7)(4,6)(5,8)(9,11,16,27)(10,17,21,36)(12,23,24,31)(13,30,32,29)(14,25,15,26)(18,34,38,39)(19,37,40,28)(20,35,33,22);(1,3)(2,5)(4,8)(6,7)(9,12,16,24)(10,18,21,38)(11,23,27,31)(13,26,32,25)(14,29,15,30)(17,34,36,39)(19,35,40,22)(20,37,33,28);(3,8)(5,7)(9,13,15,31,16,32,14,23)(10,19,33,34,21,40,20,39)(11,24,25,29,27,12,26,30)(17,18,35,28,36,38,22,37);(2,6)(5,7)(9,14,16,15)(10,20,21,33)(11,25,27,26)(12,29,24,30)(13,31,32,23)(17,35,36,22)(18,37,38,28)(19,34,40,39)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=fi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6904010);