Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573652 |
| 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 | |
2594 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)(11,17)(12,28)(13,19)(14,20)(15,22)(16,23)(18,30)(21,25)(24,37)(26,35)(27,36)(29,40)(31,38)(32,34)(33,39);(1,3,5,4)(2,6,7,8)(9,11,16,27)(10,17,23,36)(12,29,31,32)(13,25,26,15)(14,30,24,33)(18,37,39,20)(19,21,35,22)(28,40,38,34);(9,12,16,31)(10,18,23,39)(11,24,27,14)(13,29,26,32)(15,33,25,30)(17,34,36,40)(19,37,35,20)(21,38,22,28);(1,4,5,3)(2,6,7,8)(9,13)(10,19)(11,25)(12,24)(14,31)(15,27)(16,26)(17,21)(18,34)(20,38)(22,36)(23,35)(28,37)(29,33)(30,32)(39,40);(9,14)(10,20)(11,12)(13,30)(15,29)(16,24)(17,28)(18,19)(21,34)(22,40)(23,37)(25,32)(26,33)(27,31)(35,39)(36,38)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=eh,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, 6483934);