Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993899 |
| 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 | |
1807 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)(9,10,14,20,16,12,15,19)(11,21,24,22,26,33,25,17)(13,18,29,36,30,27,31,35)(23,37,28,38,39,34,40,32);(1,2)(3,7)(4,6)(5,8)(9,11,14,24,16,26,15,25)(10,17,19,33,12,22,20,21)(13,28,31,23,30,40,29,39)(18,34,36,32,27,37,35,38);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,22)(13,27)(14,20)(15,19)(17,26)(18,30)(21,24)(23,38)(25,33)(28,37)(29,36)(31,35)(32,39)(34,40);(3,8)(5,7)(9,13)(10,18)(11,23)(12,27)(14,29)(15,31)(16,30)(17,32)(19,35)(20,36)(21,37)(22,38)(24,28)(25,40)(26,39)(33,34);(2,6)(5,7)(9,14,16,15)(10,19,12,20)(11,24,26,25)(13,29,30,31)(17,33,22,21)(18,35,27,36)(23,28,39,40)(32,34,38,37)Polycyclic presentation:
2^9|a^2=g,b^2=fi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6910281);