Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573635 |
| 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 | #60374 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2578 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)(3,8)(4,6)(5,7)(9,10)(11,21)(12,13)(14,18)(15,20)(16,25)(17,30)(19,33)(22,39)(23,38)(24,27)(26,36)(28,35)(29,31)(32,34)(37,40);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,28)(13,23)(14,25)(15,26)(17,37)(18,21)(19,36)(20,24)(22,30)(27,33)(29,38)(31,35)(32,39)(34,40);(9,12,14,29)(10,17,18,32)(11,22,25,40)(13,19,31,20)(15,34,33,30)(16,35,21,23)(24,37,36,39)(26,28,27,38);(1,4,5,3)(2,6,7,8)(9,13,14,31)(10,12,18,29)(11,23,25,35)(15,32,33,17)(16,28,21,38)(19,30,20,34)(22,24,40,36)(26,39,27,37);(9,10)(11,24)(12,17)(13,30)(14,18)(15,19)(16,26)(20,33)(21,27)(22,37)(23,38)(25,36)(28,35)(29,32)(31,34)(39,40)Polycyclic presentation:
2^9|b^2=fi,c^2=i,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=ehi,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6484154);