Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996395 |
| 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 | |
2239 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)(9,10,13,20,15,21,14,19)(11,22,24,17,25,33,31,16)(12,18,28,37,30,38,29,36)(23,35,26,34,27,40,39,32);(1,2)(3,7)(4,6)(5,8)(9,11,15,25)(10,16,21,17)(12,26,30,39)(13,31,14,24)(18,35,38,40)(19,22,20,33)(23,29,27,28)(32,36,34,37);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,27)(13,24)(14,31)(15,25)(16,21)(18,32)(19,22)(20,33)(23,30)(26,29)(28,39)(34,38)(35,37)(36,40);(3,8)(5,7)(9,12,14,29,15,30,13,28)(10,18,19,36,21,38,20,37)(11,23,31,39,25,27,24,26)(16,32,33,40,17,34,22,35);(2,6)(5,7)(9,13,15,14)(10,19,21,20)(11,24,25,31)(12,28,30,29)(16,33,17,22)(18,36,38,37)(23,26,27,39)(32,40,34,35)Polycyclic presentation:
2^9|a^2=f,b^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=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6903600);