Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574063 |
| 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 | |
1985 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)(3,8)(4,6)(5,7)(9,10)(11,17)(12,18)(13,19)(14,29)(15,20)(16,21)(22,31)(23,33)(24,34)(25,32)(26,35)(27,37)(28,38)(30,36)(39,40);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,14)(13,26)(15,25)(16,23)(18,29)(19,35)(20,32)(21,33)(22,27)(24,28)(30,39)(31,37)(34,38)(36,40);(9,11,14,12,16,23,15,25)(10,18,20,17,21,32,29,33)(13,22,27,24,28,39,30,26)(19,34,36,31,38,35,37,40);(1,4,5,3)(2,6,7,8)(9,12)(10,18)(11,15)(13,22)(14,23)(16,25)(17,20)(19,31)(21,32)(24,27)(26,30)(28,39)(29,33)(34,37)(35,36)(38,40);(9,13)(10,19)(11,22)(12,24)(14,27)(15,30)(16,28)(17,31)(18,34)(20,36)(21,38)(23,39)(25,26)(29,37)(32,35)(33,40)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,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, 6491516);