Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996405 |
| 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 | |
2267 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)(9,10,14,20,16,21,15,19)(11,22,24,33,12,25,27,17)(13,18,29,36,31,37,30,35)(23,34,28,40,26,38,39,32);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,16)(13,28)(14,27)(15,24)(18,34)(19,22)(20,25)(21,33)(23,29)(26,30)(31,39)(32,36)(35,40)(37,38);(1,3)(2,5)(4,8)(6,7)(9,12,16,11)(10,17,21,33)(13,26,31,23)(14,27,15,24)(18,32,37,40)(19,25,20,22)(28,29,39,30)(34,35,38,36);(3,8)(5,7)(9,13,15,30,16,31,14,29)(10,18,19,35,21,37,20,36)(11,23,27,39,12,26,24,28)(17,32,25,38,33,40,22,34);(2,6)(5,7)(9,14,16,15)(10,19,21,20)(11,24,12,27)(13,29,31,30)(17,25,33,22)(18,35,37,36)(23,28,26,39)(32,38,40,34)Polycyclic presentation:
2^9|a^2=f,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=dhi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6903609);